Calculus Examples

$lim_{x \to 2} \frac{3 - \sqrt{2 x + 5}}{x - 2}$

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.

$\frac{lim_{x \to 2} 3 - \sqrt{2 x + 5}}{lim_{x \to 2} x - 2}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{lim_{x \to 2} 3 - lim_{x \to 2} \sqrt{2 x + 5}}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.2

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

$\frac{3 - lim_{x \to 2} \sqrt{2 x + 5}}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.3

Move the limit under the radical sign.

$\frac{3 - \sqrt{lim_{x \to 2} 2 x + 5}}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.4

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

$\frac{3 - \sqrt{lim_{x \to 2} 2 x + lim_{x \to 2} 5}}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 - \sqrt{2 lim_{x \to 2} x + lim_{x \to 2} 5}}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.6

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

$\frac{3 - \sqrt{2 lim_{x \to 2} x + 5}}{lim_{x \to 2} x - 2}$

Step 1.1.2.2

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

$\frac{3 - \sqrt{2 \cdot 2 + 5}}{lim_{x \to 2} x - 2}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Multiply $2$ by $2$ .

$\frac{3 - \sqrt{4 + 5}}{lim_{x \to 2} x - 2}$

Step 1.1.2.3.1.2

Add $4$ and $5$ .

$\frac{3 - \sqrt{9}}{lim_{x \to 2} x - 2}$

Step 1.1.2.3.1.3

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

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

Step 1.1.2.3.1.4

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

$\frac{3 - 1 \cdot 3}{lim_{x \to 2} x - 2}$

Step 1.1.2.3.1.5

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{lim_{x \to 2} x - 2}$

Step 1.1.2.3.2

Subtract $3$ from $3$ .

$\frac{0}{lim_{x \to 2} x - 2}$

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

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

$\frac{0}{lim_{x \to 2} x - lim_{x \to 2} 2}$

Step 1.1.3.1.2

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

$\frac{0}{lim_{x \to 2} x - 1 \cdot 2}$

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $2$ .

$\frac{0}{2 - 2}$

Step 1.1.3.3.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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.

$lim_{x \to 2} \frac{3 - \sqrt{2 x + 5}}{x - 2} = lim_{x \to 2} \frac{\frac{d}{d x} [3 - \sqrt{2 x + 5}]}{\frac{d}{d x} [x - 2]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 2} \frac{\frac{d}{d x} [3 - \sqrt{2 x + 5}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.2

By the Sum Rule, the derivative of $3 - \sqrt{2 x + 5}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- \sqrt{2 x + 5}]$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [3] + \frac{d}{d x} [- \sqrt{2 x + 5}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{0 + \frac{d}{d x} [- \sqrt{2 x + 5}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \sqrt{2 x + 5}]$ .

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Step 1.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 5}$ as ${(2 x + 5)}^{\frac{1}{2}}$ .

$lim_{x \to 2} \frac{0 + \frac{d}{d x} [- {(2 x + 5)}^{\frac{1}{2}}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(2 x + 5)}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$ .

$lim_{x \to 2} \frac{0 - \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.3

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 $g (x) = 2 x + 5$ .

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Step 1.3.4.3.1

To apply the Chain Rule, set $u$ as $2 x + 5$ .

$lim_{x \to 2} \frac{0 - (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [2 x + 5])}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.3.2

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

$lim_{x \to 2} \frac{0 - (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 5])}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.3.3

Replace all occurrences of $u$ with $2 x + 5$ .

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 5])}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.4

By the Sum Rule, the derivative of $2 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} (\frac{d}{d x} [2 x] + \frac{d}{d x} [5]))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.5

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

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} (2 \frac{d}{d x} [x] + \frac{d}{d x} [5]))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.6

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

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} (2 \cdot 1 + \frac{d}{d x} [5]))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.7

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} (2 \cdot 1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.8

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

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.9

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

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.10

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{1 - 1 \cdot 2}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.11

Simplify the numerator.

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Step 1.3.4.11.1

Multiply $- 1$ by $2$ .

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{1 - 2}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.11.2

Subtract $2$ from $1$ .

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{\frac{- 1}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.12

Move the negative in front of the fraction.

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{- \frac{1}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.13

Multiply $2$ by $1$ .

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{- \frac{1}{2}} (2 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.14

Add $2$ and $0$ .

$lim_{x \to 2} \frac{0 - (\frac{1}{2} {(2 x + 5)}^{- \frac{1}{2}} \cdot 2)}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.15

Combine $\frac{1}{2}$ and ${(2 x + 5)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{0 - (\frac{{(2 x + 5)}^{- \frac{1}{2}}}{2} \cdot 2)}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.16

Combine $\frac{{(2 x + 5)}^{- \frac{1}{2}}}{2}$ and $2$ .

$lim_{x \to 2} \frac{0 - \frac{{(2 x + 5)}^{- \frac{1}{2}} \cdot 2}{2}}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.17

Move $2$ to the left of ${(2 x + 5)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{0 - \frac{2 \cdot {(2 x + 5)}^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.18

Move ${(2 x + 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 2} \frac{0 - \frac{2}{2 {(2 x + 5)}^{\frac{1}{2}}}}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.19

Cancel the common factor.

$lim_{x \to 2} \frac{0 - \frac{2}{2 {(2 x + 5)}^{\frac{1}{2}}}}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.20

Rewrite the expression.

$lim_{x \to 2} \frac{0 - \frac{1}{{(2 x + 5)}^{\frac{1}{2}}}}{\frac{d}{d x} [x - 2]}$

Step 1.3.5

Subtract $\frac{1}{{(2 x + 5)}^{\frac{1}{2}}}$ from $0$ .

$lim_{x \to 2} \frac{- \frac{1}{{(2 x + 5)}^{\frac{1}{2}}}}{\frac{d}{d x} [x - 2]}$

Step 1.3.6

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

$lim_{x \to 2} \frac{- \frac{1}{{(2 x + 5)}^{\frac{1}{2}}}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 2]}$

Step 1.3.7

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

$lim_{x \to 2} \frac{- \frac{1}{{(2 x + 5)}^{\frac{1}{2}}}}{1 + \frac{d}{d x} [- 2]}$

Step 1.3.8

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

$lim_{x \to 2} \frac{- \frac{1}{{(2 x + 5)}^{\frac{1}{2}}}}{1 + 0}$

Step 1.3.9

Add $1$ and $0$ .

$lim_{x \to 2} \frac{- \frac{1}{{(2 x + 5)}^{\frac{1}{2}}}}{1}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 2} - \frac{1}{{(2 x + 5)}^{\frac{1}{2}}} \cdot 1$

Step 1.5

Rewrite ${(2 x + 5)}^{\frac{1}{2}}$ as $\sqrt{2 x + 5}$ .

$lim_{x \to 2} - \frac{1}{\sqrt{2 x + 5}} \cdot 1$

Step 1.6

Multiply $- 1$ by $1$ .

$lim_{x \to 2} - \frac{1}{\sqrt{2 x + 5}}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to 2} \frac{1}{\sqrt{2 x + 5}}$

Step 2.2

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

$- \frac{lim_{x \to 2} 1}{lim_{x \to 2} \sqrt{2 x + 5}}$

Step 2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

$- \frac{1}{lim_{x \to 2} \sqrt{2 x + 5}}$

Step 2.4

Move the limit under the radical sign.

$- \frac{1}{\sqrt{lim_{x \to 2} 2 x + 5}}$

Step 2.5

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

$- \frac{1}{\sqrt{lim_{x \to 2} 2 x + lim_{x \to 2} 5}}$

Step 2.6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{\sqrt{2 lim_{x \to 2} x + lim_{x \to 2} 5}}$

Step 2.7

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

$- \frac{1}{\sqrt{2 lim_{x \to 2} x + 5}}$

Step 3

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

$- \frac{1}{\sqrt{2 \cdot 2 + 5}}$

Step 4

Simplify the denominator.

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Step 4.1

Multiply $2$ by $2$ .

$- \frac{1}{\sqrt{4 + 5}}$

Step 4.2

Add $4$ and $5$ .

$- \frac{1}{\sqrt{9}}$

Step 4.3

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

$- \frac{1}{\sqrt{3^{2}}}$

Step 4.4

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

$- \frac{1}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{3}$

Decimal Form:

$- 0.\overline{3}$