Calculus Examples

$lim_{x \to 0} \frac{\sqrt{9 + 19 x} - \sqrt{9 + 15 x}}{3 x}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to 0} \sqrt{9 + 19 x} - \sqrt{9 + 15 x}}{lim_{x \to 0} 3 x}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \sqrt{9 + 19 x} - lim_{x \to 0} \sqrt{9 + 15 x}}{lim_{x \to 0} 3 x}$

Step 1.2.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} 9 + 19 x} - lim_{x \to 0} \sqrt{9 + 15 x}}{lim_{x \to 0} 3 x}$

Step 1.2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} 9 + lim_{x \to 0} 19 x} - lim_{x \to 0} \sqrt{9 + 15 x}}{lim_{x \to 0} 3 x}$

Step 1.2.4

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{9 + lim_{x \to 0} 19 x} - lim_{x \to 0} \sqrt{9 + 15 x}}{lim_{x \to 0} 3 x}$

Step 1.2.5

Move the term $19$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{9 + 19 lim_{x \to 0} x} - lim_{x \to 0} \sqrt{9 + 15 x}}{lim_{x \to 0} 3 x}$

Step 1.2.6

Move the limit under the radical sign.

$\frac{\sqrt{9 + 19 lim_{x \to 0} x} - \sqrt{lim_{x \to 0} 9 + 15 x}}{lim_{x \to 0} 3 x}$

Step 1.2.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sqrt{9 + 19 lim_{x \to 0} x} - \sqrt{lim_{x \to 0} 9 + lim_{x \to 0} 15 x}}{lim_{x \to 0} 3 x}$

Step 1.2.8

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{9 + 19 lim_{x \to 0} x} - \sqrt{9 + lim_{x \to 0} 15 x}}{lim_{x \to 0} 3 x}$

Step 1.2.9

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{9 + 19 lim_{x \to 0} x} - \sqrt{9 + 15 lim_{x \to 0} x}}{lim_{x \to 0} 3 x}$

Step 1.2.10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.2.10.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sqrt{9 + 19 \cdot 0} - \sqrt{9 + 15 lim_{x \to 0} x}}{lim_{x \to 0} 3 x}$

Step 1.2.10.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sqrt{9 + 19 \cdot 0} - \sqrt{9 + 15 \cdot 0}}{lim_{x \to 0} 3 x}$

Step 1.2.11

Simplify the answer.

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Step 1.2.11.1

Simplify each term.

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Step 1.2.11.1.1

Multiply $19$ by $0$ .

$\frac{\sqrt{9 + 0} - \sqrt{9 + 15 \cdot 0}}{lim_{x \to 0} 3 x}$

Step 1.2.11.1.2

Add $9$ and $0$ .

$\frac{\sqrt{9} - \sqrt{9 + 15 \cdot 0}}{lim_{x \to 0} 3 x}$

Step 1.2.11.1.3

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}} - \sqrt{9 + 15 \cdot 0}}{lim_{x \to 0} 3 x}$

Step 1.2.11.1.4

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

$\frac{3 - \sqrt{9 + 15 \cdot 0}}{lim_{x \to 0} 3 x}$

Step 1.2.11.1.5

Multiply $15$ by $0$ .

$\frac{3 - \sqrt{9 + 0}}{lim_{x \to 0} 3 x}$

Step 1.2.11.1.6

Add $9$ and $0$ .

$\frac{3 - \sqrt{9}}{lim_{x \to 0} 3 x}$

Step 1.2.11.1.7

Rewrite $9$ as $3^{2}$ .

$\frac{3 - \sqrt{3^{2}}}{lim_{x \to 0} 3 x}$

Step 1.2.11.1.8

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

$\frac{3 - 1 \cdot 3}{lim_{x \to 0} 3 x}$

Step 1.2.11.1.9

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{lim_{x \to 0} 3 x}$

Step 1.2.11.2

Subtract $3$ from $3$ .

$\frac{0}{lim_{x \to 0} 3 x}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

$\frac{0}{3 lim_{x \to 0} x}$

Step 1.3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{3 \cdot 0}$

Step 1.3.3

Multiply $3$ by $0$ .

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\sqrt{9 + 19 x} - \sqrt{9 + 15 x}}{3 x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{9 + 19 x} - \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{9 + 19 x} - \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.2

By the Sum Rule, the derivative of $\sqrt{9 + 19 x} - \sqrt{9 + 15 x}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{9 + 19 x}] + \frac{d}{d x} [- \sqrt{9 + 15 x}]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{9 + 19 x}] + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3

Evaluate $\frac{d}{d x} [\sqrt{9 + 19 x}]$ .

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + 19 x}$ as ${(9 + 19 x)}^{\frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [{(9 + 19 x)}^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 9 + 19 x$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u_{1}$ as $9 + 19 x$ .

$lim_{x \to 0} \frac{\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [9 + 19 x] + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.2.2

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

$lim_{x \to 0} \frac{\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [9 + 19 x] + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.2.3

Replace all occurrences of $u_{1}$ with $9 + 19 x$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [9 + 19 x] + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.3

By the Sum Rule, the derivative of $9 + 19 x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [19 x]$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{1}{2} - 1} (\frac{d}{d x} [9] + \frac{d}{d x} [19 x]) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.4

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [19 x]) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.5

Since $19$ is constant with respect to $x$ , the derivative of $19 x$ with respect to $x$ is $19 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{1}{2} - 1} (0 + 19 \frac{d}{d x} [x]) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.6

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

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{1}{2} - 1} (0 + 19 \cdot 1) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.7

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

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 + 19 \cdot 1) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.8

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

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 + 19 \cdot 1) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.9

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{1 - 1 \cdot 2}{2}} (0 + 19 \cdot 1) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.10

Simplify the numerator.

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Step 3.3.10.1

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{1 - 2}{2}} (0 + 19 \cdot 1) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.10.2

Subtract $2$ from $1$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{\frac{- 1}{2}} (0 + 19 \cdot 1) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.11

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{- \frac{1}{2}} (0 + 19 \cdot 1) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.12

Multiply $19$ by $1$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{- \frac{1}{2}} (0 + 19) + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.13

Add $0$ and $19$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(9 + 19 x)}^{- \frac{1}{2}} \cdot 19 + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.14

Combine $\frac{1}{2}$ and ${(9 + 19 x)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{{(9 + 19 x)}^{- \frac{1}{2}}}{2} \cdot 19 + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.15

Combine $\frac{{(9 + 19 x)}^{- \frac{1}{2}}}{2}$ and $19$ .

$lim_{x \to 0} \frac{\frac{{(9 + 19 x)}^{- \frac{1}{2}} \cdot 19}{2} + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.16

Move $19$ to the left of ${(9 + 19 x)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{19 \cdot {(9 + 19 x)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.17

Multiply $19$ by ${(9 + 19 x)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{19 {(9 + 19 x)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.3.18

Move ${(9 + 19 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} + \frac{d}{d x} [- \sqrt{9 + 15 x}]}{\frac{d}{d x} [3 x]}$

Step 3.4

Evaluate $\frac{d}{d x} [- \sqrt{9 + 15 x}]$ .

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Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + 15 x}$ as ${(9 + 15 x)}^{\frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} + \frac{d}{d x} [- {(9 + 15 x)}^{\frac{1}{2}}]}{\frac{d}{d x} [3 x]}$

Step 3.4.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(9 + 15 x)}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(9 + 15 x)}^{\frac{1}{2}}]$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - \frac{d}{d x} [{(9 + 15 x)}^{\frac{1}{2}}]}{\frac{d}{d x} [3 x]}$

Step 3.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 9 + 15 x$ .

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Step 3.4.3.1

To apply the Chain Rule, set $u_{2}$ as $9 + 15 x$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [9 + 15 x])}{\frac{d}{d x} [3 x]}$

Step 3.4.3.2

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

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [9 + 15 x])}{\frac{d}{d x} [3 x]}$

Step 3.4.3.3

Replace all occurrences of $u_{2}$ with $9 + 15 x$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [9 + 15 x])}{\frac{d}{d x} [3 x]}$

Step 3.4.4

By the Sum Rule, the derivative of $9 + 15 x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [15 x]$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{1}{2} - 1} (\frac{d}{d x} [9] + \frac{d}{d x} [15 x]))}{\frac{d}{d x} [3 x]}$

Step 3.4.5

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [15 x]))}{\frac{d}{d x} [3 x]}$

Step 3.4.6

Since $15$ is constant with respect to $x$ , the derivative of $15 x$ with respect to $x$ is $15 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{1}{2} - 1} (0 + 15 \frac{d}{d x} [x]))}{\frac{d}{d x} [3 x]}$

Step 3.4.7

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

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{1}{2} - 1} (0 + 15 \cdot 1))}{\frac{d}{d x} [3 x]}$

Step 3.4.8

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

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 + 15 \cdot 1))}{\frac{d}{d x} [3 x]}$

Step 3.4.9

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

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 + 15 \cdot 1))}{\frac{d}{d x} [3 x]}$

Step 3.4.10

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{1 - 1 \cdot 2}{2}} (0 + 15 \cdot 1))}{\frac{d}{d x} [3 x]}$

Step 3.4.11

Simplify the numerator.

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Step 3.4.11.1

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{1 - 2}{2}} (0 + 15 \cdot 1))}{\frac{d}{d x} [3 x]}$

Step 3.4.11.2

Subtract $2$ from $1$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{\frac{- 1}{2}} (0 + 15 \cdot 1))}{\frac{d}{d x} [3 x]}$

Step 3.4.12

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{- \frac{1}{2}} (0 + 15 \cdot 1))}{\frac{d}{d x} [3 x]}$

Step 3.4.13

Multiply $15$ by $1$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{- \frac{1}{2}} (0 + 15))}{\frac{d}{d x} [3 x]}$

Step 3.4.14

Add $0$ and $15$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{1}{2} {(9 + 15 x)}^{- \frac{1}{2}} \cdot 15)}{\frac{d}{d x} [3 x]}$

Step 3.4.15

Combine $\frac{1}{2}$ and ${(9 + 15 x)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - (\frac{{(9 + 15 x)}^{- \frac{1}{2}}}{2} \cdot 15)}{\frac{d}{d x} [3 x]}$

Step 3.4.16

Combine $\frac{{(9 + 15 x)}^{- \frac{1}{2}}}{2}$ and $15$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - \frac{{(9 + 15 x)}^{- \frac{1}{2}} \cdot 15}{2}}{\frac{d}{d x} [3 x]}$

Step 3.4.17

Move $15$ to the left of ${(9 + 15 x)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - \frac{15 \cdot {(9 + 15 x)}^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [3 x]}$

Step 3.4.18

Multiply $15$ by ${(9 + 15 x)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - \frac{15 {(9 + 15 x)}^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [3 x]}$

Step 3.4.19

Move ${(9 + 15 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - \frac{15}{2 {(9 + 15 x)}^{\frac{1}{2}}}}{\frac{d}{d x} [3 x]}$

Step 3.5

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - \frac{15}{2 {(9 + 15 x)}^{\frac{1}{2}}}}{3 \frac{d}{d x} [x]}$

Step 3.6

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

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - \frac{15}{2 {(9 + 15 x)}^{\frac{1}{2}}}}{3 \cdot 1}$

Step 3.7

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{\frac{19}{2 {(9 + 19 x)}^{\frac{1}{2}}} - \frac{15}{2 {(9 + 15 x)}^{\frac{1}{2}}}}{3}$

Step 4

Convert fractional exponents to radicals.

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Step 4.1

Rewrite ${(9 + 19 x)}^{\frac{1}{2}}$ as $\sqrt{9 + 19 x}$ .

$lim_{x \to 0} \frac{\frac{19}{2 \sqrt{9 + 19 x}} - \frac{15}{2 {(9 + 15 x)}^{\frac{1}{2}}}}{3}$

Step 4.2

Rewrite ${(9 + 15 x)}^{\frac{1}{2}}$ as $\sqrt{9 + 15 x}$ .

$lim_{x \to 0} \frac{\frac{19}{2 \sqrt{9 + 19 x}} - \frac{15}{2 \sqrt{9 + 15 x}}}{3}$

Step 5

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 0} \frac{19}{2 \sqrt{9 + 19 x}} - \frac{15}{2 \sqrt{9 + 15 x}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} (lim_{x \to 0} \frac{19}{2 \sqrt{9 + 19 x}} - lim_{x \to 0} \frac{15}{2 \sqrt{9 + 15 x}})$

Step 7

Move the term $\frac{19}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} (\frac{19}{2} lim_{x \to 0} \frac{1}{\sqrt{9 + 19 x}} - lim_{x \to 0} \frac{15}{2 \sqrt{9 + 15 x}})$

Step 8

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} \sqrt{9 + 19 x}} - lim_{x \to 0} \frac{15}{2 \sqrt{9 + 15 x}})$

Step 9

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{lim_{x \to 0} \sqrt{9 + 19 x}} - lim_{x \to 0} \frac{15}{2 \sqrt{9 + 15 x}})$

Step 10

Move the limit under the radical sign.

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} 9 + 19 x}} - lim_{x \to 0} \frac{15}{2 \sqrt{9 + 15 x}})$

Step 11

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} 9 + lim_{x \to 0} 19 x}} - lim_{x \to 0} \frac{15}{2 \sqrt{9 + 15 x}})$

Step 12

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + lim_{x \to 0} 19 x}} - lim_{x \to 0} \frac{15}{2 \sqrt{9 + 15 x}})$

Step 13

Move the term $19$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 lim_{x \to 0} x}} - lim_{x \to 0} \frac{15}{2 \sqrt{9 + 15 x}})$

Step 14

Move the term $\frac{15}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 lim_{x \to 0} x}} - \frac{15}{2} lim_{x \to 0} \frac{1}{\sqrt{9 + 15 x}})$

Step 15

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 lim_{x \to 0} x}} - \frac{15}{2} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} \sqrt{9 + 15 x}})$

Step 16

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 lim_{x \to 0} x}} - \frac{15}{2} \cdot \frac{1}{lim_{x \to 0} \sqrt{9 + 15 x}})$

Step 17

Move the limit under the radical sign.

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 lim_{x \to 0} x}} - \frac{15}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} 9 + 15 x}})$

Step 18

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 lim_{x \to 0} x}} - \frac{15}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} 9 + lim_{x \to 0} 15 x}})$

Step 19

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 lim_{x \to 0} x}} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + lim_{x \to 0} 15 x}})$

Step 20

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 lim_{x \to 0} x}} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 15 lim_{x \to 0} x}})$

Step 21

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 21.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 \cdot 0}} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 15 lim_{x \to 0} x}})$

Step 21.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 19 \cdot 0}} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 15 \cdot 0}})$

Step 22

Simplify the answer.

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Step 22.1

Simplify each term.

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Step 22.1.1

Simplify the denominator.

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Step 22.1.1.1

Multiply $19$ by $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9 + 0}} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 15 \cdot 0}})$

Step 22.1.1.2

Add $9$ and $0$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{9}} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 15 \cdot 0}})$

Step 22.1.1.3

Rewrite $9$ as $3^{2}$ .

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{\sqrt{3^{2}}} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 15 \cdot 0}})$

Step 22.1.1.4

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

$\frac{1}{3} (\frac{19}{2} \cdot \frac{1}{3} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 15 \cdot 0}})$

Step 22.1.2

Multiply $\frac{19}{2} \cdot \frac{1}{3}$ .

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Step 22.1.2.1

Multiply $\frac{19}{2}$ by $\frac{1}{3}$ .

$\frac{1}{3} (\frac{19}{2 \cdot 3} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 15 \cdot 0}})$

Step 22.1.2.2

Multiply $2$ by $3$ .

$\frac{1}{3} (\frac{19}{6} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 15 \cdot 0}})$

Step 22.1.3

Simplify the denominator.

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Step 22.1.3.1

Multiply $15$ by $0$ .

$\frac{1}{3} (\frac{19}{6} - \frac{15}{2} \cdot \frac{1}{\sqrt{9 + 0}})$

Step 22.1.3.2

Add $9$ and $0$ .

$\frac{1}{3} (\frac{19}{6} - \frac{15}{2} \cdot \frac{1}{\sqrt{9}})$

Step 22.1.3.3

Rewrite $9$ as $3^{2}$ .

$\frac{1}{3} (\frac{19}{6} - \frac{15}{2} \cdot \frac{1}{\sqrt{3^{2}}})$

Step 22.1.3.4

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

$\frac{1}{3} (\frac{19}{6} - \frac{15}{2} \cdot \frac{1}{3})$

Step 22.1.4

Cancel the common factor of $3$ .

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Step 22.1.4.1

Move the leading negative in $- \frac{15}{2}$ into the numerator.

$\frac{1}{3} (\frac{19}{6} + \frac{- 15}{2} \cdot \frac{1}{3})$

Step 22.1.4.2

Factor $3$ out of $- 15$ .

$\frac{1}{3} (\frac{19}{6} + \frac{3 (- 5)}{2} \cdot \frac{1}{3})$

Step 22.1.4.3

Cancel the common factor.

$\frac{1}{3} (\frac{19}{6} + \frac{3 \cdot - 5}{2} \cdot \frac{1}{3})$

Step 22.1.4.4

Rewrite the expression.

$\frac{1}{3} (\frac{19}{6} + \frac{- 5}{2})$

Step 22.1.5

Move the negative in front of the fraction.

$\frac{1}{3} (\frac{19}{6} - \frac{5}{2})$

Step 22.2

To write $- \frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} (\frac{19}{6} - \frac{5}{2} \cdot \frac{3}{3})$

Step 22.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 22.3.1

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

$\frac{1}{3} (\frac{19}{6} - \frac{5 \cdot 3}{2 \cdot 3})$

Step 22.3.2

Multiply $2$ by $3$ .

$\frac{1}{3} (\frac{19}{6} - \frac{5 \cdot 3}{6})$

Step 22.4

Combine the numerators over the common denominator.

$\frac{1}{3} \cdot \frac{19 - 5 \cdot 3}{6}$

Step 22.5

Simplify the numerator.

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Step 22.5.1

Multiply $- 5$ by $3$ .

$\frac{1}{3} \cdot \frac{19 - 15}{6}$

Step 22.5.2

Subtract $15$ from $19$ .

$\frac{1}{3} \cdot \frac{4}{6}$

Step 22.6

Cancel the common factor of $4$ and $6$ .

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Step 22.6.1

Factor $2$ out of $4$ .

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

Step 22.6.2

Cancel the common factors.

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Step 22.6.2.1

Factor $2$ out of $6$ .

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

Step 22.6.2.2

Cancel the common factor.

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

Step 22.6.2.3

Rewrite the expression.

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

Step 22.7

Multiply $\frac{1}{3} \cdot \frac{2}{3}$ .

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Step 22.7.1

Multiply $\frac{1}{3}$ by $\frac{2}{3}$ .

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

Step 22.7.2

Multiply $3$ by $3$ .

$\frac{2}{9}$