Calculus Examples

$lim_{t \to 2} \frac{\sqrt{(t + 4) {(t - 2)}^{4}}}{{(3 t - 6)}^{2}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{t \to 2} \sqrt{(t + 4) {(t - 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{t \to 2} (t + 4) {(t - 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.2

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{\sqrt{lim_{t \to 2} t + 4 \cdot lim_{t \to 2} {(t - 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{\sqrt{(lim_{t \to 2} t + lim_{t \to 2} 4) \cdot lim_{t \to 2} {(t - 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.4

Evaluate the limit of $4$ which is constant as $t$ approaches $2$ .

$\frac{\sqrt{(lim_{t \to 2} t + 4) \cdot lim_{t \to 2} {(t - 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.5

Move the exponent $4$ from ${(t - 2)}^{4}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{(lim_{t \to 2} t + 4) \cdot {(lim_{t \to 2} t - 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.6

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{\sqrt{(lim_{t \to 2} t + 4) \cdot {(lim_{t \to 2} t - lim_{t \to 2} 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.7

Evaluate the limit of $2$ which is constant as $t$ approaches $2$ .

$\frac{\sqrt{(lim_{t \to 2} t + 4) \cdot {(lim_{t \to 2} t - 1 \cdot 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.8

Evaluate the limits by plugging in $2$ for all occurrences of $t$ .

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Step 1.1.2.8.1

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{\sqrt{(2 + 4) \cdot {(lim_{t \to 2} t - 1 \cdot 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.8.2

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{\sqrt{(2 + 4) \cdot {(2 - 1 \cdot 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.9

Simplify the answer.

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Step 1.1.2.9.1

Add $2$ and $4$ .

$\frac{\sqrt{6 {(2 - 1 \cdot 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.9.2

Multiply $- 1$ by $2$ .

$\frac{\sqrt{6 {(2 - 2)}^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.9.3

Subtract $2$ from $2$ .

$\frac{\sqrt{6 \cdot 0^{4}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.9.4

Raising $0$ to any positive power yields $0$ .

$\frac{\sqrt{6 \cdot 0}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.9.5

Multiply $6$ by $0$ .

$\frac{\sqrt{0}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.9.6

Rewrite $0$ as $0^{2}$ .

$\frac{\sqrt{0^{2}}}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.2.9.7

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0}{lim_{t \to 2} {(3 t - 6)}^{2}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Move the exponent $2$ from ${(3 t - 6)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{t \to 2} 3 t - 6)}^{2}}$

Step 1.1.3.1.2

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{0}{{(lim_{t \to 2} 3 t - lim_{t \to 2} 6)}^{2}}$

Step 1.1.3.1.3

Move the term $3$ outside of the limit because it is constant with respect to $t$ .

$\frac{0}{{(3 lim_{t \to 2} t - lim_{t \to 2} 6)}^{2}}$

Step 1.1.3.1.4

Evaluate the limit of $6$ which is constant as $t$ approaches $2$ .

$\frac{0}{{(3 lim_{t \to 2} t - 1 \cdot 6)}^{2}}$

Step 1.1.3.2

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{0}{{(3 \cdot 2 - 1 \cdot 6)}^{2}}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Multiply $3$ by $2$ .

$\frac{0}{{(6 - 1 \cdot 6)}^{2}}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $6$ .

$\frac{0}{{(6 - 6)}^{2}}$

Step 1.1.3.3.2

Subtract $6$ from $6$ .

$\frac{0}{0^{2}}$

Step 1.1.3.3.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0}$

Step 1.1.3.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{t \to 2} \frac{\sqrt{(t + 4) {(t - 2)}^{4}}}{{(3 t - 6)}^{2}} = lim_{t \to 2} \frac{\frac{d}{d t} [\sqrt{(t + 4) {(t - 2)}^{4}}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{t \to 2} \frac{\frac{d}{d t} [\sqrt{(t + 4) {(t - 2)}^{4}}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.2

Rewrite $\frac{d}{d t} [\sqrt{(t + 4) {(t - 2)}^{4}}]$ as $\frac{d}{d t} [{(t - 2)}^{2} \sqrt{t + 2^{2}}]$ .

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Step 1.3.2.1

Rewrite $(t + 4) {(t - 2)}^{4}$ as ${({(t - 2)}^{2})}^{2} (t + 2^{2})$ .

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Step 1.3.2.1.1

Rewrite ${(t - 2)}^{4}$ as ${({(t - 2)}^{2})}^{2}$ .

$lim_{t \to 2} \frac{\frac{d}{d t} [\sqrt{(t + 4) {({(t - 2)}^{2})}^{2}}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.2.1.2

Reorder $t + 4$ and ${({(t - 2)}^{2})}^{2}$ .

$lim_{t \to 2} \frac{\frac{d}{d t} [\sqrt{{({(t - 2)}^{2})}^{2} (t + 4)}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.2.1.3

Rewrite $4$ as $2^{2}$ .

$lim_{t \to 2} \frac{\frac{d}{d t} [\sqrt{{({(t - 2)}^{2})}^{2} (t + 2^{2})}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.2.2

Pull terms out from under the radical.

$lim_{t \to 2} \frac{\frac{d}{d t} [{(t - 2)}^{2} \sqrt{t + 2^{2}}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.3

Raise $2$ to the power of $2$ .

$lim_{t \to 2} \frac{\frac{d}{d t} [{(t - 2)}^{2} \sqrt{t + 4}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t + 4}$ as ${(t + 4)}^{\frac{1}{2}}$ .

$lim_{t \to 2} \frac{\frac{d}{d t} [{(t - 2)}^{2} {(t + 4)}^{\frac{1}{2}}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.5

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = {(t - 2)}^{2}$ and $g (t) = {(t + 4)}^{\frac{1}{2}}$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} \frac{d}{d t} [{(t + 4)}^{\frac{1}{2}}] + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.6

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = t + 4$ .

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Step 1.3.6.1

To apply the Chain Rule, set $u_{1}$ as $t + 4$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.6.3

Replace all occurrences of $u_{1}$ with $t + 4$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{1}{2} {(t + 4)}^{\frac{1}{2} - 1} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{1}{2} {(t + 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.8

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{1}{2} {(t + 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.9

Combine the numerators over the common denominator.

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{1}{2} {(t + 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.10

Simplify the numerator.

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Step 1.3.10.1

Multiply $- 1$ by $2$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{1}{2} {(t + 4)}^{\frac{1 - 2}{2}} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.10.2

Subtract $2$ from $1$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{1}{2} {(t + 4)}^{\frac{- 1}{2}} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.11

Move the negative in front of the fraction.

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{1}{2} {(t + 4)}^{- \frac{1}{2}} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.12

Combine $\frac{1}{2}$ and ${(t + 4)}^{- \frac{1}{2}}$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{{(t + 4)}^{- \frac{1}{2}}}{2} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.13

Move ${(t + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{t \to 2} \frac{{(t - 2)}^{2} (\frac{1}{2 {(t + 4)}^{\frac{1}{2}}} \frac{d}{d t} [t + 4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.14

Combine $\frac{1}{2 {(t + 4)}^{\frac{1}{2}}}$ and ${(t - 2)}^{2}$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} \frac{d}{d t} [t + 4] + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.15

By the Sum Rule, the derivative of $t + 4$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [4]$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} (\frac{d}{d t} [t] + \frac{d}{d t} [4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.16

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} (1 + \frac{d}{d t} [4]) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.17

Since $4$ is constant with respect to $t$ , the derivative of $4$ with respect to $t$ is $0$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} (1 + 0) + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.18

Add $1$ and $0$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} \cdot 1 + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.19

Multiply $\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}}$ by $1$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + {(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [{(t - 2)}^{2}]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.20

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = t - 2$ .

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Step 1.3.20.1

To apply the Chain Rule, set $u_{2}$ as $t - 2$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + {(t + 4)}^{\frac{1}{2}} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d t} [t - 2])}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.20.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + {(t + 4)}^{\frac{1}{2}} (2 u_{2} \frac{d}{d t} [t - 2])}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.20.3

Replace all occurrences of $u_{2}$ with $t - 2$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + {(t + 4)}^{\frac{1}{2}} (2 (t - 2) \frac{d}{d t} [t - 2])}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.21

Move $2$ to the left of ${(t + 4)}^{\frac{1}{2}}$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + 2 \cdot {(t + 4)}^{\frac{1}{2}} (t - 2) \frac{d}{d t} [t - 2]}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.22

By the Sum Rule, the derivative of $t - 2$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 2]$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + 2 {(t + 4)}^{\frac{1}{2}} (t - 2) (\frac{d}{d t} [t] + \frac{d}{d t} [- 2])}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.23

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + 2 {(t + 4)}^{\frac{1}{2}} (t - 2) (1 + \frac{d}{d t} [- 2])}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.24

Since $- 2$ is constant with respect to $t$ , the derivative of $- 2$ with respect to $t$ is $0$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + 2 {(t + 4)}^{\frac{1}{2}} (t - 2) (1 + 0)}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.25

Add $1$ and $0$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + 2 {(t + 4)}^{\frac{1}{2}} (t - 2) \cdot 1}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.26

Multiply $2$ by $1$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + 2 {(t + 4)}^{\frac{1}{2}} (t - 2)}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.27

To write $2 {(t + 4)}^{\frac{1}{2}} (t - 2)$ as a fraction with a common denominator, multiply by $\frac{2 {(t + 4)}^{\frac{1}{2}}}{2 {(t + 4)}^{\frac{1}{2}}}$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + 2 {(t + 4)}^{\frac{1}{2}} (t - 2) \cdot \frac{2 {(t + 4)}^{\frac{1}{2}}}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.28

Combine $2 {(t + 4)}^{\frac{1}{2}} (t - 2)$ and $\frac{2 {(t + 4)}^{\frac{1}{2}}}{2 {(t + 4)}^{\frac{1}{2}}}$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2}}{2 {(t + 4)}^{\frac{1}{2}}} + \frac{2 {(t + 4)}^{\frac{1}{2}} (t - 2) (2 {(t + 4)}^{\frac{1}{2}})}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.29

Combine the numerators over the common denominator.

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2} + 2 {(t + 4)}^{\frac{1}{2}} (t - 2) (2 {(t + 4)}^{\frac{1}{2}})}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.30

Multiply $2$ by $2$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2} + 4 {(t + 4)}^{\frac{1}{2}} (t - 2) {(t + 4)}^{\frac{1}{2}}}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.31

Multiply ${(t + 4)}^{\frac{1}{2}}$ by ${(t + 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.3.31.1

Move ${(t + 4)}^{\frac{1}{2}}$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2} + 4 ({(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}}) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.31.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2} + 4 {(t + 4)}^{\frac{1}{2} + \frac{1}{2}} (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.31.3

Combine the numerators over the common denominator.

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2} + 4 {(t + 4)}^{\frac{1 + 1}{2}} (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.31.4

Add $1$ and $1$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2} + 4 {(t + 4)}^{\frac{2}{2}} (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.31.5

Divide $2$ by $2$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2} + 4 {(t + 4)}^{1} (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.32

Simplify $4 {(t + 4)}^{1} (t - 2)$ .

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2} + 4 (t + 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33

Simplify.

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Step 1.3.33.1

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{{(t - 2)}^{2} + (4 t + 4 \cdot 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2

Simplify the numerator.

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Step 1.3.33.2.1

Simplify each term.

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Step 1.3.33.2.1.1

Rewrite ${(t - 2)}^{2}$ as $(t - 2) (t - 2)$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (t - 2) + (4 t + 4 \cdot 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.2

Expand $(t - 2) (t - 2)$ using the FOIL Method.

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Step 1.3.33.2.1.2.1

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{t (t - 2) - 2 (t - 2) + (4 t + 4 \cdot 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.2.2

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{t \cdot t + t \cdot - 2 - 2 (t - 2) + (4 t + 4 \cdot 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.2.3

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{t \cdot t + t \cdot - 2 - 2 t - 2 \cdot - 2 + (4 t + 4 \cdot 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.3

Simplify and combine like terms.

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Step 1.3.33.2.1.3.1

Simplify each term.

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Step 1.3.33.2.1.3.1.1

Multiply $t$ by $t$ .

$lim_{t \to 2} \frac{\frac{t^{2} + t \cdot - 2 - 2 t - 2 \cdot - 2 + (4 t + 4 \cdot 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.3.1.2

Move $- 2$ to the left of $t$ .

$lim_{t \to 2} \frac{\frac{t^{2} - 2 \cdot t - 2 t - 2 \cdot - 2 + (4 t + 4 \cdot 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.3.1.3

Multiply $- 2$ by $- 2$ .

$lim_{t \to 2} \frac{\frac{t^{2} - 2 t - 2 t + 4 + (4 t + 4 \cdot 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.3.2

Subtract $2 t$ from $- 2 t$ .

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + (4 t + 4 \cdot 4) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.4

Multiply $4$ by $4$ .

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + (4 t + 16) (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.5

Expand $(4 t + 16) (t - 2)$ using the FOIL Method.

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Step 1.3.33.2.1.5.1

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + 4 t (t - 2) + 16 (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.5.2

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + 4 t \cdot t + 4 t \cdot - 2 + 16 (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.5.3

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + 4 t \cdot t + 4 t \cdot - 2 + 16 t + 16 \cdot - 2}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.6

Simplify and combine like terms.

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Step 1.3.33.2.1.6.1

Simplify each term.

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Step 1.3.33.2.1.6.1.1

Multiply $t$ by $t$ by adding the exponents.

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Step 1.3.33.2.1.6.1.1.1

Move $t$ .

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + 4 (t \cdot t) + 4 t \cdot - 2 + 16 t + 16 \cdot - 2}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.6.1.1.2

Multiply $t$ by $t$ .

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + 4 t^{2} + 4 t \cdot - 2 + 16 t + 16 \cdot - 2}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.6.1.2

Multiply $- 2$ by $4$ .

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + 4 t^{2} - 8 t + 16 t + 16 \cdot - 2}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.6.1.3

Multiply $16$ by $- 2$ .

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + 4 t^{2} - 8 t + 16 t - 32}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.1.6.2

Add $- 8 t$ and $16 t$ .

$lim_{t \to 2} \frac{\frac{t^{2} - 4 t + 4 + 4 t^{2} + 8 t - 32}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.2

Add $t^{2}$ and $4 t^{2}$ .

$lim_{t \to 2} \frac{\frac{5 t^{2} - 4 t + 4 + 8 t - 32}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.3

Add $- 4 t$ and $8 t$ .

$lim_{t \to 2} \frac{\frac{5 t^{2} + 4 t + 4 - 32}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.2.4

Subtract $32$ from $4$ .

$lim_{t \to 2} \frac{\frac{5 t^{2} + 4 t - 28}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.3

Factor by grouping.

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Step 1.3.33.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 28 = - 140$ and whose sum is $b = 4$ .

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Step 1.3.33.3.1.1

Factor $4$ out of $4 t$ .

$lim_{t \to 2} \frac{\frac{5 t^{2} + 4 (t) - 28}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.3.1.2

Rewrite $4$ as $- 10$ plus $14$

$lim_{t \to 2} \frac{\frac{5 t^{2} + (- 10 + 14) t - 28}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.3.1.3

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{5 t^{2} - 10 t + 14 t - 28}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.3.2

Factor out the greatest common factor from each group.

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Step 1.3.33.3.2.1

Group the first two terms and the last two terms.

$lim_{t \to 2} \frac{\frac{(5 t^{2} - 10 t) + 14 t - 28}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.3.2.2

Factor out the greatest common factor (GCF) from each group.

$lim_{t \to 2} \frac{\frac{5 t (t - 2) + 14 (t - 2)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.33.3.3

Factor the polynomial by factoring out the greatest common factor, $t - 2$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [{(3 t - 6)}^{2}]}$

Step 1.3.34

Rewrite ${(3 t - 6)}^{2}$ as $(3 t - 6) (3 t - 6)$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [(3 t - 6) (3 t - 6)]}$

Step 1.3.35

Expand $(3 t - 6) (3 t - 6)$ using the FOIL Method.

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Step 1.3.35.1

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [3 t (3 t - 6) - 6 (3 t - 6)]}$

Step 1.3.35.2

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [3 t (3 t) + 3 t \cdot - 6 - 6 (3 t - 6)]}$

Step 1.3.35.3

Apply the distributive property.

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [3 t (3 t) + 3 t \cdot - 6 - 6 (3 t) - 6 \cdot - 6]}$

Step 1.3.36

Simplify and combine like terms.

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Step 1.3.36.1

Simplify each term.

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Step 1.3.36.1.1

Rewrite using the commutative property of multiplication.

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [3 \cdot 3 t \cdot t + 3 t \cdot - 6 - 6 (3 t) - 6 \cdot - 6]}$

Step 1.3.36.1.2

Multiply $t$ by $t$ by adding the exponents.

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Step 1.3.36.1.2.1

Move $t$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [3 \cdot 3 (t \cdot t) + 3 t \cdot - 6 - 6 (3 t) - 6 \cdot - 6]}$

Step 1.3.36.1.2.2

Multiply $t$ by $t$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [3 \cdot 3 t^{2} + 3 t \cdot - 6 - 6 (3 t) - 6 \cdot - 6]}$

Step 1.3.36.1.3

Multiply $3$ by $3$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [9 t^{2} + 3 t \cdot - 6 - 6 (3 t) - 6 \cdot - 6]}$

Step 1.3.36.1.4

Multiply $- 6$ by $3$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [9 t^{2} - 18 t - 6 (3 t) - 6 \cdot - 6]}$

Step 1.3.36.1.5

Multiply $3$ by $- 6$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [9 t^{2} - 18 t - 18 t - 6 \cdot - 6]}$

Step 1.3.36.1.6

Multiply $- 6$ by $- 6$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [9 t^{2} - 18 t - 18 t + 36]}$

Step 1.3.36.2

Subtract $18 t$ from $- 18 t$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [9 t^{2} - 36 t + 36]}$

Step 1.3.37

By the Sum Rule, the derivative of $9 t^{2} - 36 t + 36$ with respect to $t$ is $\frac{d}{d t} [9 t^{2}] + \frac{d}{d t} [- 36 t] + \frac{d}{d t} [36]$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{\frac{d}{d t} [9 t^{2}] + \frac{d}{d t} [- 36 t] + \frac{d}{d t} [36]}$

Step 1.3.38

Since $9$ is constant with respect to $t$ , the derivative of $9 t^{2}$ with respect to $t$ is $9 \frac{d}{d t} [t^{2}]$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{9 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 36 t] + \frac{d}{d t} [36]}$

Step 1.3.39

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{9 (2 t) + \frac{d}{d t} [- 36 t] + \frac{d}{d t} [36]}$

Step 1.3.40

Multiply $2$ by $9$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{18 t + \frac{d}{d t} [- 36 t] + \frac{d}{d t} [36]}$

Step 1.3.41

Since $- 36$ is constant with respect to $t$ , the derivative of $- 36 t$ with respect to $t$ is $- 36 \frac{d}{d t} [t]$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{18 t - 36 \frac{d}{d t} [t] + \frac{d}{d t} [36]}$

Step 1.3.42

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{18 t - 36 \cdot 1 + \frac{d}{d t} [36]}$

Step 1.3.43

Multiply $- 36$ by $1$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{18 t - 36 + \frac{d}{d t} [36]}$

Step 1.3.44

Since $36$ is constant with respect to $t$ , the derivative of $36$ with respect to $t$ is $0$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{18 t - 36 + 0}$

Step 1.3.45

Add $18 t - 36$ and $0$ .

$lim_{t \to 2} \frac{\frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}}}{18 t - 36}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{t \to 2} \frac{(t - 2) (5 t + 14)}{2 {(t + 4)}^{\frac{1}{2}}} \cdot \frac{1}{18 t - 36}$

Step 1.5

Rewrite ${(t + 4)}^{\frac{1}{2}}$ as $\sqrt{t + 4}$ .

$lim_{t \to 2} \frac{(t - 2) (5 t + 14)}{2 \sqrt{t + 4}} \cdot \frac{1}{18 t - 36}$

Step 1.6

Multiply $\frac{(t - 2) (5 t + 14)}{2 \sqrt{t + 4}}$ by $\frac{1}{18 t - 36}$ .

$lim_{t \to 2} \frac{(t - 2) (5 t + 14)}{2 \sqrt{t + 4} (18 t - 36)}$

Step 2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $t$ .

$\frac{1}{2} lim_{t \to 2} \frac{(t - 2) (5 t + 14)}{\sqrt{t + 4} (18 t - 36)}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{1}{2} \cdot \frac{lim_{t \to 2} (t - 2) (5 t + 14)}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{lim_{t \to 2} t - 2 \cdot lim_{t \to 2} 5 t + 14}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.2

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{(lim_{t \to 2} t - lim_{t \to 2} 2) \cdot lim_{t \to 2} 5 t + 14}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.3

Evaluate the limit of $2$ which is constant as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{(lim_{t \to 2} t - 1 \cdot 2) \cdot lim_{t \to 2} 5 t + 14}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.4

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{(lim_{t \to 2} t - 1 \cdot 2) \cdot (lim_{t \to 2} 5 t + lim_{t \to 2} 14)}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.5

Move the term $5$ outside of the limit because it is constant with respect to $t$ .

$\frac{1}{2} \cdot \frac{(lim_{t \to 2} t - 1 \cdot 2) \cdot (5 lim_{t \to 2} t + lim_{t \to 2} 14)}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.6

Evaluate the limit of $14$ which is constant as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{(lim_{t \to 2} t - 1 \cdot 2) \cdot (5 lim_{t \to 2} t + 14)}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.7

Evaluate the limits by plugging in $2$ for all occurrences of $t$ .

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Step 3.1.2.7.1

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{1}{2} \cdot \frac{(2 - 1 \cdot 2) \cdot (5 lim_{t \to 2} t + 14)}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.7.2

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{1}{2} \cdot \frac{(2 - 1 \cdot 2) \cdot (5 \cdot 2 + 14)}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.8

Simplify the answer.

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Step 3.1.2.8.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} \cdot \frac{(2 - 2) \cdot (5 \cdot 2 + 14)}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.8.2

Subtract $2$ from $2$ .

$\frac{1}{2} \cdot \frac{0 \cdot (5 \cdot 2 + 14)}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.8.3

Multiply $5$ by $2$ .

$\frac{1}{2} \cdot \frac{0 \cdot (10 + 14)}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.8.4

Add $10$ and $14$ .

$\frac{1}{2} \cdot \frac{0 \cdot 24}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.2.8.5

Multiply $0$ by $24$ .

$\frac{1}{2} \cdot \frac{0}{lim_{t \to 2} \sqrt{t + 4} (18 t - 36)}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{0}{lim_{t \to 2} \sqrt{t + 4} \cdot lim_{t \to 2} 18 t - 36}$

Step 3.1.3.2

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{0}{\sqrt{lim_{t \to 2} t + 4} \cdot lim_{t \to 2} 18 t - 36}$

Step 3.1.3.3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{lim_{t \to 2} t + lim_{t \to 2} 4} \cdot lim_{t \to 2} 18 t - 36}$

Step 3.1.3.4

Evaluate the limit of $4$ which is constant as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{lim_{t \to 2} t + 4} \cdot lim_{t \to 2} 18 t - 36}$

Step 3.1.3.5

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{lim_{t \to 2} t + 4} \cdot (lim_{t \to 2} 18 t - lim_{t \to 2} 36)}$

Step 3.1.3.6

Move the term $18$ outside of the limit because it is constant with respect to $t$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{lim_{t \to 2} t + 4} \cdot (18 lim_{t \to 2} t - lim_{t \to 2} 36)}$

Step 3.1.3.7

Evaluate the limit of $36$ which is constant as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{lim_{t \to 2} t + 4} \cdot (18 lim_{t \to 2} t - 1 \cdot 36)}$

Step 3.1.3.8

Evaluate the limits by plugging in $2$ for all occurrences of $t$ .

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Step 3.1.3.8.1

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{2 + 4} \cdot (18 lim_{t \to 2} t - 1 \cdot 36)}$

Step 3.1.3.8.2

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{2 + 4} \cdot (18 \cdot 2 - 1 \cdot 36)}$

Step 3.1.3.9

Simplify the answer.

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Step 3.1.3.9.1

Add $2$ and $4$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{6} \cdot (18 \cdot 2 - 1 \cdot 36)}$

Step 3.1.3.9.2

Simplify each term.

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Step 3.1.3.9.2.1

Multiply $18$ by $2$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{6} \cdot (36 - 1 \cdot 36)}$

Step 3.1.3.9.2.2

Multiply $- 1$ by $36$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{6} \cdot (36 - 36)}$

Step 3.1.3.9.3

Subtract $36$ from $36$ .

$\frac{1}{2} \cdot \frac{0}{\sqrt{6} \cdot 0}$

Step 3.1.3.9.4

Multiply $\sqrt{6}$ by $0$ .

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.3.9.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.3.10

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{t \to 2} \frac{(t - 2) (5 t + 14)}{\sqrt{t + 4} (18 t - 36)} = lim_{t \to 2} \frac{\frac{d}{d t} [(t - 2) (5 t + 14)]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$\frac{1}{2} lim_{t \to 2} \frac{\frac{d}{d t} [(t - 2) (5 t + 14)]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t - 2$ and $g (t) = 5 t + 14$ .

$\frac{1}{2} lim_{t \to 2} \frac{(t - 2) \frac{d}{d t} [5 t + 14] + (5 t + 14) \frac{d}{d t} [t - 2]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.3

By the Sum Rule, the derivative of $5 t + 14$ with respect to $t$ is $\frac{d}{d t} [5 t] + \frac{d}{d t} [14]$ .

$\frac{1}{2} lim_{t \to 2} \frac{(t - 2) (\frac{d}{d t} [5 t] + \frac{d}{d t} [14]) + (5 t + 14) \frac{d}{d t} [t - 2]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.4

Since $5$ is constant with respect to $t$ , the derivative of $5 t$ with respect to $t$ is $5 \frac{d}{d t} [t]$ .

$\frac{1}{2} lim_{t \to 2} \frac{(t - 2) (5 \frac{d}{d t} [t] + \frac{d}{d t} [14]) + (5 t + 14) \frac{d}{d t} [t - 2]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.5

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{1}{2} lim_{t \to 2} \frac{(t - 2) (5 \cdot 1 + \frac{d}{d t} [14]) + (5 t + 14) \frac{d}{d t} [t - 2]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.6

Multiply $5$ by $1$ .

$\frac{1}{2} lim_{t \to 2} \frac{(t - 2) (5 + \frac{d}{d t} [14]) + (5 t + 14) \frac{d}{d t} [t - 2]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.7

Since $14$ is constant with respect to $t$ , the derivative of $14$ with respect to $t$ is $0$ .

$\frac{1}{2} lim_{t \to 2} \frac{(t - 2) (5 + 0) + (5 t + 14) \frac{d}{d t} [t - 2]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.8

Add $5$ and $0$ .

$\frac{1}{2} lim_{t \to 2} \frac{(t - 2) \cdot 5 + (5 t + 14) \frac{d}{d t} [t - 2]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.9

Move $5$ to the left of $t - 2$ .

$\frac{1}{2} lim_{t \to 2} \frac{5 \cdot (t - 2) + (5 t + 14) \frac{d}{d t} [t - 2]}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.10

By the Sum Rule, the derivative of $t - 2$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 2]$ .

$\frac{1}{2} lim_{t \to 2} \frac{5 (t - 2) + (5 t + 14) (\frac{d}{d t} [t] + \frac{d}{d t} [- 2])}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.11

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{1}{2} lim_{t \to 2} \frac{5 (t - 2) + (5 t + 14) (1 + \frac{d}{d t} [- 2])}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.12

Since $- 2$ is constant with respect to $t$ , the derivative of $- 2$ with respect to $t$ is $0$ .

$\frac{1}{2} lim_{t \to 2} \frac{5 (t - 2) + (5 t + 14) (1 + 0)}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.13

Add $1$ and $0$ .

$\frac{1}{2} lim_{t \to 2} \frac{5 (t - 2) + (5 t + 14) \cdot 1}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.14

Multiply $5 t + 14$ by $1$ .

$\frac{1}{2} lim_{t \to 2} \frac{5 (t - 2) + 5 t + 14}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.15

Simplify.

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Step 3.3.15.1

Apply the distributive property.

$\frac{1}{2} lim_{t \to 2} \frac{5 t + 5 \cdot - 2 + 5 t + 14}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.15.2

Combine terms.

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Step 3.3.15.2.1

Multiply $5$ by $- 2$ .

$\frac{1}{2} lim_{t \to 2} \frac{5 t - 10 + 5 t + 14}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.15.2.2

Add $5 t$ and $5 t$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t - 10 + 14}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.15.2.3

Add $- 10$ and $14$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{d}{d t} [\sqrt{t + 4} (18 t - 36)]}$

Step 3.3.16

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t + 4}$ as ${(t + 4)}^{\frac{1}{2}}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{d}{d t} [{(t + 4)}^{\frac{1}{2}} (18 t - 36)]}$

Step 3.3.17

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = {(t + 4)}^{\frac{1}{2}}$ and $g (t) = 18 t - 36$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{{(t + 4)}^{\frac{1}{2}} \frac{d}{d t} [18 t - 36] + (18 t - 36) \frac{d}{d t} [{(t + 4)}^{\frac{1}{2}}]}$

Step 3.3.18

By the Sum Rule, the derivative of $18 t - 36$ with respect to $t$ is $\frac{d}{d t} [18 t] + \frac{d}{d t} [- 36]$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{{(t + 4)}^{\frac{1}{2}} (\frac{d}{d t} [18 t] + \frac{d}{d t} [- 36]) + (18 t - 36) \frac{d}{d t} [{(t + 4)}^{\frac{1}{2}}]}$

Step 3.3.19

Since $18$ is constant with respect to $t$ , the derivative of $18 t$ with respect to $t$ is $18 \frac{d}{d t} [t]$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{{(t + 4)}^{\frac{1}{2}} (18 \frac{d}{d t} [t] + \frac{d}{d t} [- 36]) + (18 t - 36) \frac{d}{d t} [{(t + 4)}^{\frac{1}{2}}]}$

Step 3.3.20

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{{(t + 4)}^{\frac{1}{2}} (18 \cdot 1 + \frac{d}{d t} [- 36]) + (18 t - 36) \frac{d}{d t} [{(t + 4)}^{\frac{1}{2}}]}$

Step 3.3.21

Multiply $18$ by $1$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{{(t + 4)}^{\frac{1}{2}} (18 + \frac{d}{d t} [- 36]) + (18 t - 36) \frac{d}{d t} [{(t + 4)}^{\frac{1}{2}}]}$

Step 3.3.22

Since $- 36$ is constant with respect to $t$ , the derivative of $- 36$ with respect to $t$ is $0$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{{(t + 4)}^{\frac{1}{2}} (18 + 0) + (18 t - 36) \frac{d}{d t} [{(t + 4)}^{\frac{1}{2}}]}$

Step 3.3.23

Add $18$ and $0$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{{(t + 4)}^{\frac{1}{2}} \cdot 18 + (18 t - 36) \frac{d}{d t} [{(t + 4)}^{\frac{1}{2}}]}$

Step 3.3.24

Move $18$ to the left of ${(t + 4)}^{\frac{1}{2}}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 \cdot {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{d}{d t} [{(t + 4)}^{\frac{1}{2}}]}$

Step 3.3.25

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = t + 4$ .

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Step 3.3.25.1

To apply the Chain Rule, set $u$ as $t + 4$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d t} [t + 4]}$

Step 3.3.25.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d t} [t + 4]}$

Step 3.3.25.3

Replace all occurrences of $u$ with $t + 4$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2} {(t + 4)}^{\frac{1}{2} - 1} \frac{d}{d t} [t + 4]}$

Step 3.3.26

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2} {(t + 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} [t + 4]}$

Step 3.3.27

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2} {(t + 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} [t + 4]}$

Step 3.3.28

Combine the numerators over the common denominator.

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2} {(t + 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} [t + 4]}$

Step 3.3.29

Simplify the numerator.

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Step 3.3.29.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2} {(t + 4)}^{\frac{1 - 2}{2}} \frac{d}{d t} [t + 4]}$

Step 3.3.29.2

Subtract $2$ from $1$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2} {(t + 4)}^{\frac{- 1}{2}} \frac{d}{d t} [t + 4]}$

Step 3.3.30

Move the negative in front of the fraction.

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2} {(t + 4)}^{- \frac{1}{2}} \frac{d}{d t} [t + 4]}$

Step 3.3.31

Combine $\frac{1}{2}$ and ${(t + 4)}^{- \frac{1}{2}}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{{(t + 4)}^{- \frac{1}{2}}}{2} \frac{d}{d t} [t + 4]}$

Step 3.3.32

Move ${(t + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2 {(t + 4)}^{\frac{1}{2}}} \frac{d}{d t} [t + 4]}$

Step 3.3.33

By the Sum Rule, the derivative of $t + 4$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [4]$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2 {(t + 4)}^{\frac{1}{2}}} (\frac{d}{d t} [t] + \frac{d}{d t} [4])}$

Step 3.3.34

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2 {(t + 4)}^{\frac{1}{2}}} (1 + \frac{d}{d t} [4])}$

Step 3.3.35

Since $4$ is constant with respect to $t$ , the derivative of $4$ with respect to $t$ is $0$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2 {(t + 4)}^{\frac{1}{2}}} (1 + 0)}$

Step 3.3.36

Add $1$ and $0$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2 {(t + 4)}^{\frac{1}{2}}} \cdot 1}$

Step 3.3.37

Multiply $18 t - 36$ by $1$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{18 {(t + 4)}^{\frac{1}{2}} + (18 t - 36) \frac{1}{2 {(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38

Simplify.

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Step 3.3.38.1

Reorder terms.

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{(18 t - 36) \frac{1}{2 {(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2

Simplify each term.

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Step 3.3.38.2.1

Multiply $18 t - 36$ by $\frac{1}{2 {(t + 4)}^{\frac{1}{2}}}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{18 t - 36}{2 {(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2.2

Factor $2$ out of $18 t$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{2 \cdot 9 t - 36}{2 {(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2.3

Factor $2$ out of $- 36$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{2 \cdot 9 t + 2 \cdot - 18}{2 {(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2.4

Factor $2$ out of $2 (9 t) + 2 (- 18)$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{2 (9 t - 18)}{2 {(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2.5

Cancel the common factors.

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Step 3.3.38.2.5.1

Factor $2$ out of $2 {(t + 4)}^{\frac{1}{2}}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{2 (9 t - 18)}{2 ({(t + 4)}^{\frac{1}{2}})} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2.5.2

Cancel the common factor.

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{2 (9 t - 18)}{2 {(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2.5.3

Rewrite the expression.

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 t - 18}{{(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2.6

Factor $9$ out of $9 t - 18$ .

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Step 3.3.38.2.6.1

Factor $9$ out of $9 t$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t) - 18}{{(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2.6.2

Factor $9$ out of $- 18$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 t + 9 \cdot - 2}{{(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.2.6.3

Factor $9$ out of $9 t + 9 \cdot - 2$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2)}{{(t + 4)}^{\frac{1}{2}}} + 18 {(t + 4)}^{\frac{1}{2}}}$

Step 3.3.38.3

To write $18 {(t + 4)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(t + 4)}^{\frac{1}{2}}}{{(t + 4)}^{\frac{1}{2}}}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2)}{{(t + 4)}^{\frac{1}{2}}} + \frac{18 {(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}}}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.4

Combine the numerators over the common denominator.

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2) + 18 {(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}}}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5

Simplify the numerator.

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Step 3.3.38.5.1

Factor $9$ out of $9 (t - 2) + 18 {(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}}$ .

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Step 3.3.38.5.1.1

Factor $9$ out of $18 {(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2) + 9 \cdot 2 {(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}}}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.1.2

Factor $9$ out of $9 (t - 2) + 9 (2 {(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}})$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2 + 2 {(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}})}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.2

Multiply ${(t + 4)}^{\frac{1}{2}}$ by ${(t + 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 3.3.38.5.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2 + 2 {(t + 4)}^{\frac{1}{2} + \frac{1}{2}})}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.2.2

Combine the numerators over the common denominator.

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2 + 2 {(t + 4)}^{\frac{1 + 1}{2}})}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.2.3

Add $1$ and $1$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2 + 2 {(t + 4)}^{\frac{2}{2}})}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.2.4

Divide $2$ by $2$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2 + 2 {(t + 4)}^{1})}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.3

Simplify $2 {(t + 4)}^{1}$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2 + 2 (t + 4))}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.4

Apply the distributive property.

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2 + 2 t + 2 \cdot 4)}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.5

Multiply $2$ by $4$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (t - 2 + 2 t + 8)}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.6

Add $t$ and $2 t$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (3 t - 2 + 8)}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.7

Add $- 2$ and $8$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (3 t + 6)}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.8

Factor $3$ out of $3 t + 6$ .

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Step 3.3.38.5.8.1

Factor $3$ out of $3 t$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (3 (t) + 6)}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.8.2

Factor $3$ out of $6$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 (3 t + 3 \cdot 2)}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.8.3

Factor $3$ out of $3 t + 3 \cdot 2$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{9 \cdot 3 (t + 2)}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.3.38.5.9

Multiply $9$ by $3$ .

$\frac{1}{2} lim_{t \to 2} \frac{10 t + 4}{\frac{27 (t + 2)}{{(t + 4)}^{\frac{1}{2}}}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} lim_{t \to 2} (10 t + 4) \frac{{(t + 4)}^{\frac{1}{2}}}{27 (t + 2)}$

Step 3.5

Rewrite ${(t + 4)}^{\frac{1}{2}}$ as $\sqrt{t + 4}$ .

$\frac{1}{2} lim_{t \to 2} (10 t + 4) \frac{\sqrt{t + 4}}{27 (t + 2)}$

Step 3.6

Multiply $10 t + 4$ by $\frac{\sqrt{t + 4}}{27 (t + 2)}$ .

$\frac{1}{2} lim_{t \to 2} \frac{(10 t + 4) \sqrt{t + 4}}{27 (t + 2)}$

Step 4

Evaluate the limit.

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Step 4.1

Move the term $\frac{1}{27}$ outside of the limit because it is constant with respect to $t$ .

$\frac{1}{2} \cdot \frac{1}{27} lim_{t \to 2} \frac{(10 t + 4) \sqrt{t + 4}}{t + 2}$

Step 4.2

Split the limit using the Limits Quotient Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{lim_{t \to 2} (10 t + 4) \sqrt{t + 4}}{lim_{t \to 2} t + 2}$

Step 4.3

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{lim_{t \to 2} 10 t + 4 \cdot lim_{t \to 2} \sqrt{t + 4}}{lim_{t \to 2} t + 2}$

Step 4.4

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(lim_{t \to 2} 10 t + lim_{t \to 2} 4) \cdot lim_{t \to 2} \sqrt{t + 4}}{lim_{t \to 2} t + 2}$

Step 4.5

Move the term $10$ outside of the limit because it is constant with respect to $t$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 lim_{t \to 2} t + lim_{t \to 2} 4) \cdot lim_{t \to 2} \sqrt{t + 4}}{lim_{t \to 2} t + 2}$

Step 4.6

Evaluate the limit of $4$ which is constant as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 lim_{t \to 2} t + 4) \cdot lim_{t \to 2} \sqrt{t + 4}}{lim_{t \to 2} t + 2}$

Step 4.7

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 lim_{t \to 2} t + 4) \cdot \sqrt{lim_{t \to 2} t + 4}}{lim_{t \to 2} t + 2}$

Step 4.8

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 lim_{t \to 2} t + 4) \cdot \sqrt{lim_{t \to 2} t + lim_{t \to 2} 4}}{lim_{t \to 2} t + 2}$

Step 4.9

Evaluate the limit of $4$ which is constant as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 lim_{t \to 2} t + 4) \cdot \sqrt{lim_{t \to 2} t + 4}}{lim_{t \to 2} t + 2}$

Step 4.10

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 lim_{t \to 2} t + 4) \cdot \sqrt{lim_{t \to 2} t + 4}}{lim_{t \to 2} t + lim_{t \to 2} 2}$

Step 4.11

Evaluate the limit of $2$ which is constant as $t$ approaches $2$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 lim_{t \to 2} t + 4) \cdot \sqrt{lim_{t \to 2} t + 4}}{lim_{t \to 2} t + 2}$

Step 5

Evaluate the limits by plugging in $2$ for all occurrences of $t$ .

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Step 5.1

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 \cdot 2 + 4) \cdot \sqrt{lim_{t \to 2} t + 4}}{lim_{t \to 2} t + 2}$

Step 5.2

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 \cdot 2 + 4) \cdot \sqrt{2 + 4}}{lim_{t \to 2} t + 2}$

Step 5.3

Evaluate the limit of $t$ by plugging in $2$ for $t$ .

$\frac{1}{2} \cdot \frac{1}{27} \frac{(10 \cdot 2 + 4) \cdot \sqrt{2 + 4}}{2 + 2}$

Step 6

Simplify the answer.

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Step 6.1

Multiply $\frac{1}{2} \cdot \frac{1}{27}$ .

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Step 6.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{27}$ .

$\frac{1}{2 \cdot 27} \cdot \frac{(10 \cdot 2 + 4) \cdot \sqrt{2 + 4}}{2 + 2}$

Step 6.1.2

Multiply $2$ by $27$ .

$\frac{1}{54} \cdot \frac{(10 \cdot 2 + 4) \cdot \sqrt{2 + 4}}{2 + 2}$

Step 6.2

Cancel the common factor of $10 \cdot 2 + 4$ and $2 + 2$ .

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Step 6.2.1

Factor $2$ out of $(10 \cdot 2 + 4) \cdot \sqrt{2 + 4}$ .

$\frac{1}{54} \cdot \frac{2 ((5 \cdot 2 + 2) \cdot \sqrt{2 + 4})}{2 + 2}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $2$ out of $2$ .

$\frac{1}{54} \cdot \frac{2 ((5 \cdot 2 + 2) \cdot \sqrt{2 + 4})}{2 \cdot 1 + 2}$

Step 6.2.2.2

Factor $2$ out of $2$ .

$\frac{1}{54} \cdot \frac{2 ((5 \cdot 2 + 2) \cdot \sqrt{2 + 4})}{2 \cdot 1 + 2 (1)}$

Step 6.2.2.3

Factor $2$ out of $2 \cdot 1 + 2 (1)$ .

$\frac{1}{54} \cdot \frac{2 ((5 \cdot 2 + 2) \cdot \sqrt{2 + 4})}{2 \cdot (1 + 1)}$

Step 6.2.2.4

Cancel the common factor.

$\frac{1}{54} \cdot \frac{2 ((5 \cdot 2 + 2) \cdot \sqrt{2 + 4})}{2 \cdot (1 + 1)}$

Step 6.2.2.5

Rewrite the expression.

$\frac{1}{54} \cdot \frac{(5 \cdot 2 + 2) \cdot \sqrt{2 + 4}}{1 + 1}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Multiply $5$ by $2$ .

$\frac{1}{54} \cdot \frac{(10 + 2) \sqrt{2 + 4}}{1 + 1}$

Step 6.3.2

Add $10$ and $2$ .

$\frac{1}{54} \cdot \frac{12 \sqrt{2 + 4}}{1 + 1}$

Step 6.3.3

Add $2$ and $4$ .

$\frac{1}{54} \cdot \frac{12 \sqrt{6}}{1 + 1}$

Step 6.4

Add $1$ and $1$ .

$\frac{1}{54} \cdot \frac{12 \sqrt{6}}{2}$

Step 6.5

Cancel the common factor of $6$ .

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Step 6.5.1

Factor $6$ out of $54$ .

$\frac{1}{6 (9)} \cdot \frac{12 \sqrt{6}}{2}$

Step 6.5.2

Factor $6$ out of $12 \sqrt{6}$ .

$\frac{1}{6 (9)} \cdot \frac{6 (2 \sqrt{6})}{2}$

Step 6.5.3

Cancel the common factor.

$\frac{1}{6 \cdot 9} \cdot \frac{6 (2 \sqrt{6})}{2}$

Step 6.5.4

Rewrite the expression.

$\frac{1}{9} \cdot \frac{2 \sqrt{6}}{2}$

Step 6.6

Multiply $\frac{1}{9}$ by $\frac{2 \sqrt{6}}{2}$ .

$\frac{2 \sqrt{6}}{9 \cdot 2}$

Step 6.7

Multiply $9$ by $2$ .

$\frac{2 \sqrt{6}}{18}$

Step 6.8

Cancel the common factor of $2$ and $18$ .

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Step 6.8.1

Factor $2$ out of $2 \sqrt{6}$ .

$\frac{2 (\sqrt{6})}{18}$

Step 6.8.2

Cancel the common factors.

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Step 6.8.2.1

Factor $2$ out of $18$ .

$\frac{2 \sqrt{6}}{2 \cdot 9}$

Step 6.8.2.2

Cancel the common factor.

$\frac{2 \sqrt{6}}{2 \cdot 9}$

Step 6.8.2.3

Rewrite the expression.

$\frac{\sqrt{6}}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{9}$

Decimal Form:

$0.27216552 \dots$