Calculus Examples

$lim_{v \to 8} \frac{v - 2 - \sqrt{v^{2} - 28}}{v - 8}$

Step 1

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

Tap for more steps...

Step 1.1

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

$\frac{lim_{v \to 8} v - 2 - \sqrt{v^{2} - 28}}{lim_{v \to 8} v - 8}$

Step 1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as $v$ approaches $8$ .

$\frac{lim_{v \to 8} v - lim_{v \to 8} 2 - lim_{v \to 8} \sqrt{v^{2} - 28}}{lim_{v \to 8} v - 8}$

Step 1.2.2

Evaluate the limit of $2$ which is constant as $v$ approaches $8$ .

$\frac{lim_{v \to 8} v - 1 \cdot 2 - lim_{v \to 8} \sqrt{v^{2} - 28}}{lim_{v \to 8} v - 8}$

Step 1.2.3

Move the limit under the radical sign.

$\frac{lim_{v \to 8} v - 1 \cdot 2 - \sqrt{lim_{v \to 8} v^{2} - 28}}{lim_{v \to 8} v - 8}$

Step 1.2.4

Split the limit using the Sum of Limits Rule on the limit as $v$ approaches $8$ .

$\frac{lim_{v \to 8} v - 1 \cdot 2 - \sqrt{lim_{v \to 8} v^{2} - lim_{v \to 8} 28}}{lim_{v \to 8} v - 8}$

Step 1.2.5

Move the exponent $2$ from $v^{2}$ outside the limit using the Limits Power Rule.

$\frac{lim_{v \to 8} v - 1 \cdot 2 - \sqrt{{(lim_{v \to 8} v)}^{2} - lim_{v \to 8} 28}}{lim_{v \to 8} v - 8}$

Step 1.2.6

Evaluate the limit of $28$ which is constant as $v$ approaches $8$ .

$\frac{lim_{v \to 8} v - 1 \cdot 2 - \sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}{lim_{v \to 8} v - 8}$

Step 1.2.7

Evaluate the limits by plugging in $8$ for all occurrences of $v$ .

Tap for more steps...

Step 1.2.7.1

Evaluate the limit of $v$ by plugging in $8$ for $v$ .

$\frac{8 - 1 \cdot 2 - \sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}{lim_{v \to 8} v - 8}$

Step 1.2.7.2

Evaluate the limit of $v$ by plugging in $8$ for $v$ .

$\frac{8 - 1 \cdot 2 - \sqrt{8^{2} - 1 \cdot 28}}{lim_{v \to 8} v - 8}$

Step 1.2.8

Simplify the answer.

Tap for more steps...

Step 1.2.8.1

Simplify each term.

Tap for more steps...

Step 1.2.8.1.1

Multiply $- 1$ by $2$ .

$\frac{8 - 2 - \sqrt{8^{2} - 1 \cdot 28}}{lim_{v \to 8} v - 8}$

Step 1.2.8.1.2

Raise $8$ to the power of $2$ .

$\frac{8 - 2 - \sqrt{64 - 1 \cdot 28}}{lim_{v \to 8} v - 8}$

Step 1.2.8.1.3

Multiply $- 1$ by $28$ .

$\frac{8 - 2 - \sqrt{64 - 28}}{lim_{v \to 8} v - 8}$

Step 1.2.8.1.4

Subtract $28$ from $64$ .

$\frac{8 - 2 - \sqrt{36}}{lim_{v \to 8} v - 8}$

Step 1.2.8.1.5

Rewrite $36$ as $6^{2}$ .

$\frac{8 - 2 - \sqrt{6^{2}}}{lim_{v \to 8} v - 8}$

Step 1.2.8.1.6

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

$\frac{8 - 2 - 1 \cdot 6}{lim_{v \to 8} v - 8}$

Step 1.2.8.1.7

Multiply $- 1$ by $6$ .

$\frac{8 - 2 - 6}{lim_{v \to 8} v - 8}$

Step 1.2.8.2

Subtract $2$ from $8$ .

$\frac{6 - 6}{lim_{v \to 8} v - 8}$

Step 1.2.8.3

Subtract $6$ from $6$ .

$\frac{0}{lim_{v \to 8} v - 8}$

Step 1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 1.3.1

Evaluate the limit.

Tap for more steps...

Step 1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $v$ approaches $8$ .

$\frac{0}{lim_{v \to 8} v - lim_{v \to 8} 8}$

Step 1.3.1.2

Evaluate the limit of $8$ which is constant as $v$ approaches $8$ .

$\frac{0}{lim_{v \to 8} v - 1 \cdot 8}$

Step 1.3.2

Evaluate the limit of $v$ by plugging in $8$ for $v$ .

$\frac{0}{8 - 1 \cdot 8}$

Step 1.3.3

Simplify the answer.

Tap for more steps...

Step 1.3.3.1

Multiply $- 1$ by $8$ .

$\frac{0}{8 - 8}$

Step 1.3.3.2

Subtract $8$ from $8$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{v \to 8} \frac{v - 2 - \sqrt{v^{2} - 28}}{v - 8} = lim_{v \to 8} \frac{\frac{d}{d v} [v - 2 - \sqrt{v^{2} - 28}]}{\frac{d}{d v} [v - 8]}$

Step 3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.1

Differentiate the numerator and denominator.

$lim_{v \to 8} \frac{\frac{d}{d v} [v - 2 - \sqrt{v^{2} - 28}]}{\frac{d}{d v} [v - 8]}$

Step 3.2

By the Sum Rule, the derivative of $v - 2 - \sqrt{v^{2} - 28}$ with respect to $v$ is $\frac{d}{d v} [v] + \frac{d}{d v} [- 2] + \frac{d}{d v} [- \sqrt{v^{2} - 28}]$ .

$lim_{v \to 8} \frac{\frac{d}{d v} [v] + \frac{d}{d v} [- 2] + \frac{d}{d v} [- \sqrt{v^{2} - 28}]}{\frac{d}{d v} [v - 8]}$

Step 3.3

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

$lim_{v \to 8} \frac{1 + \frac{d}{d v} [- 2] + \frac{d}{d v} [- \sqrt{v^{2} - 28}]}{\frac{d}{d v} [v - 8]}$

Step 3.4

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

$lim_{v \to 8} \frac{1 + 0 + \frac{d}{d v} [- \sqrt{v^{2} - 28}]}{\frac{d}{d v} [v - 8]}$

Step 3.5

Evaluate $\frac{d}{d v} [- \sqrt{v^{2} - 28}]$ .

Tap for more steps...

Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{v^{2} - 28}$ as ${(v^{2} - 28)}^{\frac{1}{2}}$ .

$lim_{v \to 8} \frac{1 + 0 + \frac{d}{d v} [- {(v^{2} - 28)}^{\frac{1}{2}}]}{\frac{d}{d v} [v - 8]}$

Step 3.5.2

Since $- 1$ is constant with respect to $v$ , the derivative of $- {(v^{2} - 28)}^{\frac{1}{2}}$ with respect to $v$ is $- \frac{d}{d v} [{(v^{2} - 28)}^{\frac{1}{2}}]$ .

$lim_{v \to 8} \frac{1 + 0 - \frac{d}{d v} [{(v^{2} - 28)}^{\frac{1}{2}}]}{\frac{d}{d v} [v - 8]}$

Step 3.5.3

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (v))]$ is $f' (g (v)) g' (v)$ where $f (v) = v^{\frac{1}{2}}$ and $g (v) = v^{2} - 28$ .

Tap for more steps...

Step 3.5.3.1

To apply the Chain Rule, set $u$ as $v^{2} - 28$ .

$lim_{v \to 8} \frac{1 + 0 - (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d v} [v^{2} - 28])}{\frac{d}{d v} [v - 8]}$

Step 3.5.3.2

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

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d v} [v^{2} - 28])}{\frac{d}{d v} [v - 8]}$

Step 3.5.3.3

Replace all occurrences of $u$ with $v^{2} - 28$ .

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{\frac{1}{2} - 1} \frac{d}{d v} [v^{2} - 28])}{\frac{d}{d v} [v - 8]}$

Step 3.5.4

By the Sum Rule, the derivative of $v^{2} - 28$ with respect to $v$ is $\frac{d}{d v} [v^{2}] + \frac{d}{d v} [- 28]$ .

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{\frac{1}{2} - 1} (\frac{d}{d v} [v^{2}] + \frac{d}{d v} [- 28]))}{\frac{d}{d v} [v - 8]}$

Step 3.5.5

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

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{\frac{1}{2} - 1} (2 v + \frac{d}{d v} [- 28]))}{\frac{d}{d v} [v - 8]}$

Step 3.5.6

Since $- 28$ is constant with respect to $v$ , the derivative of $- 28$ with respect to $v$ is $0$ .

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{\frac{1}{2} - 1} (2 v + 0))}{\frac{d}{d v} [v - 8]}$

Step 3.5.7

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

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (2 v + 0))}{\frac{d}{d v} [v - 8]}$

Step 3.5.8

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

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (2 v + 0))}{\frac{d}{d v} [v - 8]}$

Step 3.5.9

Combine the numerators over the common denominator.

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{\frac{1 - 1 \cdot 2}{2}} (2 v + 0))}{\frac{d}{d v} [v - 8]}$

Step 3.5.10

Simplify the numerator.

Tap for more steps...

Step 3.5.10.1

Multiply $- 1$ by $2$ .

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{\frac{1 - 2}{2}} (2 v + 0))}{\frac{d}{d v} [v - 8]}$

Step 3.5.10.2

Subtract $2$ from $1$ .

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{\frac{- 1}{2}} (2 v + 0))}{\frac{d}{d v} [v - 8]}$

Step 3.5.11

Move the negative in front of the fraction.

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{- \frac{1}{2}} (2 v + 0))}{\frac{d}{d v} [v - 8]}$

Step 3.5.12

Add $2 v$ and $0$ .

$lim_{v \to 8} \frac{1 + 0 - (\frac{1}{2} {(v^{2} - 28)}^{- \frac{1}{2}} (2 v))}{\frac{d}{d v} [v - 8]}$

Step 3.5.13

Combine $\frac{1}{2}$ and ${(v^{2} - 28)}^{- \frac{1}{2}}$ .

$lim_{v \to 8} \frac{1 + 0 - (\frac{{(v^{2} - 28)}^{- \frac{1}{2}}}{2} (2 v))}{\frac{d}{d v} [v - 8]}$

Step 3.5.14

Combine $2$ and $\frac{{(v^{2} - 28)}^{- \frac{1}{2}}}{2}$ .

$lim_{v \to 8} \frac{1 + 0 - (\frac{2 {(v^{2} - 28)}^{- \frac{1}{2}}}{2} v)}{\frac{d}{d v} [v - 8]}$

Step 3.5.15

Combine $\frac{2 {(v^{2} - 28)}^{- \frac{1}{2}}}{2}$ and $v$ .

$lim_{v \to 8} \frac{1 + 0 - \frac{2 {(v^{2} - 28)}^{- \frac{1}{2}} v}{2}}{\frac{d}{d v} [v - 8]}$

Step 3.5.16

Move ${(v^{2} - 28)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{v \to 8} \frac{1 + 0 - \frac{2 v}{2 {(v^{2} - 28)}^{\frac{1}{2}}}}{\frac{d}{d v} [v - 8]}$

Step 3.5.17

Cancel the common factor.

$lim_{v \to 8} \frac{1 + 0 - \frac{2 v}{2 {(v^{2} - 28)}^{\frac{1}{2}}}}{\frac{d}{d v} [v - 8]}$

Step 3.5.18

Rewrite the expression.

$lim_{v \to 8} \frac{1 + 0 - \frac{v}{{(v^{2} - 28)}^{\frac{1}{2}}}}{\frac{d}{d v} [v - 8]}$

Step 3.6

Simplify.

Tap for more steps...

Step 3.6.1

Add $1$ and $0$ .

$lim_{v \to 8} \frac{1 - \frac{v}{{(v^{2} - 28)}^{\frac{1}{2}}}}{\frac{d}{d v} [v - 8]}$

Step 3.6.2

Reorder terms.

$lim_{v \to 8} \frac{- \frac{v}{{(v^{2} - 28)}^{\frac{1}{2}}} + 1}{\frac{d}{d v} [v - 8]}$

Step 3.7

By the Sum Rule, the derivative of $v - 8$ with respect to $v$ is $\frac{d}{d v} [v] + \frac{d}{d v} [- 8]$ .

$lim_{v \to 8} \frac{- \frac{v}{{(v^{2} - 28)}^{\frac{1}{2}}} + 1}{\frac{d}{d v} [v] + \frac{d}{d v} [- 8]}$

Step 3.8

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

$lim_{v \to 8} \frac{- \frac{v}{{(v^{2} - 28)}^{\frac{1}{2}}} + 1}{1 + \frac{d}{d v} [- 8]}$

Step 3.9

Since $- 8$ is constant with respect to $v$ , the derivative of $- 8$ with respect to $v$ is $0$ .

$lim_{v \to 8} \frac{- \frac{v}{{(v^{2} - 28)}^{\frac{1}{2}}} + 1}{1 + 0}$

Step 3.10

Add $1$ and $0$ .

$lim_{v \to 8} \frac{- \frac{v}{{(v^{2} - 28)}^{\frac{1}{2}}} + 1}{1}$

Step 4

Rewrite ${(v^{2} - 28)}^{\frac{1}{2}}$ as $\sqrt{v^{2} - 28}$ .

$lim_{v \to 8} \frac{- \frac{v}{\sqrt{v^{2} - 28}} + 1}{1}$

Step 5

Combine terms.

Tap for more steps...

Step 5.1

Write $1$ as a fraction with a common denominator.

$lim_{v \to 8} \frac{- \frac{v}{\sqrt{v^{2} - 28}} + \frac{\sqrt{v^{2} - 28}}{\sqrt{v^{2} - 28}}}{1}$

Step 5.2

Combine the numerators over the common denominator.

$lim_{v \to 8} \frac{\frac{- v + \sqrt{v^{2} - 28}}{\sqrt{v^{2} - 28}}}{1}$

Step 6

Divide $\frac{- v + \sqrt{v^{2} - 28}}{\sqrt{v^{2} - 28}}$ by $1$ .

$lim_{v \to 8} \frac{- v + \sqrt{v^{2} - 28}}{\sqrt{v^{2} - 28}}$

Step 7

Split the limit using the Limits Quotient Rule on the limit as $v$ approaches $8$ .

$\frac{lim_{v \to 8} - v + \sqrt{v^{2} - 28}}{lim_{v \to 8} \sqrt{v^{2} - 28}}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $v$ approaches $8$ .

$\frac{- lim_{v \to 8} v + lim_{v \to 8} \sqrt{v^{2} - 28}}{lim_{v \to 8} \sqrt{v^{2} - 28}}$

Step 9

Move the limit under the radical sign.

$\frac{- lim_{v \to 8} v + \sqrt{lim_{v \to 8} v^{2} - 28}}{lim_{v \to 8} \sqrt{v^{2} - 28}}$

Step 10

Split the limit using the Sum of Limits Rule on the limit as $v$ approaches $8$ .

$\frac{- lim_{v \to 8} v + \sqrt{lim_{v \to 8} v^{2} - lim_{v \to 8} 28}}{lim_{v \to 8} \sqrt{v^{2} - 28}}$

Step 11

Move the exponent $2$ from $v^{2}$ outside the limit using the Limits Power Rule.

$\frac{- lim_{v \to 8} v + \sqrt{{(lim_{v \to 8} v)}^{2} - lim_{v \to 8} 28}}{lim_{v \to 8} \sqrt{v^{2} - 28}}$

Step 12

Evaluate the limit of $28$ which is constant as $v$ approaches $8$ .

$\frac{- lim_{v \to 8} v + \sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}{lim_{v \to 8} \sqrt{v^{2} - 28}}$

Step 13

Move the limit under the radical sign.

$\frac{- lim_{v \to 8} v + \sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}{\sqrt{lim_{v \to 8} v^{2} - 28}}$

Step 14

Split the limit using the Sum of Limits Rule on the limit as $v$ approaches $8$ .

$\frac{- lim_{v \to 8} v + \sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}{\sqrt{lim_{v \to 8} v^{2} - lim_{v \to 8} 28}}$

Step 15

Move the exponent $2$ from $v^{2}$ outside the limit using the Limits Power Rule.

$\frac{- lim_{v \to 8} v + \sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}{\sqrt{{(lim_{v \to 8} v)}^{2} - lim_{v \to 8} 28}}$

Step 16

Evaluate the limit of $28$ which is constant as $v$ approaches $8$ .

$\frac{- lim_{v \to 8} v + \sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}{\sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}$

Step 17

Evaluate the limits by plugging in $8$ for all occurrences of $v$ .

Tap for more steps...

Step 17.1

Evaluate the limit of $v$ by plugging in $8$ for $v$ .

$\frac{- 8 + \sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}{\sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}$

Step 17.2

Evaluate the limit of $v$ by plugging in $8$ for $v$ .

$\frac{- 8 + \sqrt{8^{2} - 1 \cdot 28}}{\sqrt{{(lim_{v \to 8} v)}^{2} - 1 \cdot 28}}$

Step 17.3

Evaluate the limit of $v$ by plugging in $8$ for $v$ .

$\frac{- 8 + \sqrt{8^{2} - 1 \cdot 28}}{\sqrt{8^{2} - 1 \cdot 28}}$

Step 18

Simplify the answer.

Tap for more steps...

Step 18.1

Simplify the numerator.

Tap for more steps...

Step 18.1.1

Raise $8$ to the power of $2$ .

$\frac{- 8 + \sqrt{64 - 1 \cdot 28}}{\sqrt{8^{2} - 1 \cdot 28}}$

Step 18.1.2

Multiply $- 1$ by $28$ .

$\frac{- 8 + \sqrt{64 - 28}}{\sqrt{8^{2} - 1 \cdot 28}}$

Step 18.1.3

Subtract $28$ from $64$ .

$\frac{- 8 + \sqrt{36}}{\sqrt{8^{2} - 1 \cdot 28}}$

Step 18.1.4

Rewrite $36$ as $6^{2}$ .

$\frac{- 8 + \sqrt{6^{2}}}{\sqrt{8^{2} - 1 \cdot 28}}$

Step 18.1.5

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

$\frac{- 8 + 6}{\sqrt{8^{2} - 1 \cdot 28}}$

Step 18.1.6

Add $- 8$ and $6$ .

$\frac{- 2}{\sqrt{8^{2} - 1 \cdot 28}}$

Step 18.2

Simplify the denominator.

Tap for more steps...

Step 18.2.1

Raise $8$ to the power of $2$ .

$\frac{- 2}{\sqrt{64 - 1 \cdot 28}}$

Step 18.2.2

Multiply $- 1$ by $28$ .

$\frac{- 2}{\sqrt{64 - 28}}$

Step 18.2.3

Subtract $28$ from $64$ .

$\frac{- 2}{\sqrt{36}}$

Step 18.2.4

Rewrite $36$ as $6^{2}$ .

$\frac{- 2}{\sqrt{6^{2}}}$

Step 18.2.5

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

$\frac{- 2}{6}$

Step 18.3

Cancel the common factor of $- 2$ and $6$ .

Tap for more steps...

Step 18.3.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{6}$

Step 18.3.2

Cancel the common factors.

Tap for more steps...

Step 18.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot - 1}{2 \cdot 3}$

Step 18.3.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 3}$

Step 18.3.2.3

Rewrite the expression.

$\frac{- 1}{3}$

Step 18.4

Move the negative in front of the fraction.

$- \frac{1}{3}$