Calculus Examples

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

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.

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

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

Move the limit under the radical sign.

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

$\frac{\sqrt{{(lim_{x \to 2^{+}} x)}^{2} - lim_{x \to 2^{+}} 4}}{lim_{x \to 2^{+}} x - 2}$

Step 1.1.2.1.4

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

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

Step 1.1.2.2

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

$\frac{\sqrt{2^{2} - 1 \cdot 4}}{lim_{x \to 2^{+}} x - 2}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

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

$\frac{\sqrt{4 - 1 \cdot 4}}{lim_{x \to 2^{+}} x - 2}$

Step 1.1.2.3.2

Multiply $- 1$ by $4$ .

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

Step 1.1.2.3.3

Subtract $4$ from $4$ .

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

Step 1.1.2.3.4

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

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

Step 1.1.2.3.5

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

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

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.

$lim_{x \to 2^{+}} \frac{\sqrt{x^{2} - 4}}{x - 2} = lim_{x \to 2^{+}} \frac{\frac{d}{d x} [\sqrt{x^{2} - 4}]}{\frac{d}{d x} [x - 2]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 2^{+}} \frac{\frac{d}{d x} [\sqrt{x^{2} - 4}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.2

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

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

Step 1.3.3

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 $g (x) = x^{2} - 4$ .

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Step 1.3.3.1

To apply the Chain Rule, set $u$ as $x^{2} - 4$ .

$lim_{x \to 2^{+}} \frac{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} - 4]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.2

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

$lim_{x \to 2^{+}} \frac{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} - 4]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.3

Replace all occurrences of $u$ with $x^{2} - 4$ .

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

Combine the numerators over the common denominator.

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

Step 1.3.7

Simplify the numerator.

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Step 1.3.7.1

Multiply $- 1$ by $2$ .

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

Step 1.3.7.2

Subtract $2$ from $1$ .

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

Step 1.3.8

Move the negative in front of the fraction.

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

Step 1.3.9

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

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

Step 1.3.10

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

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

Step 1.3.11

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

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

Step 1.3.12

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

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

Step 1.3.13

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

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

Step 1.3.14

Add $2 x$ and $0$ .

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

Step 1.3.15

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

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

Step 1.3.16

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

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

Step 1.3.17

Cancel the common factor.

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

Step 1.3.18

Rewrite the expression.

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

Step 1.3.19

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

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

Step 1.3.20

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

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

Step 1.3.21

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

$lim_{x \to 2^{+}} \frac{\frac{x}{{(x^{2} - 4)}^{\frac{1}{2}}}}{1 + 0}$

Step 1.3.22

Add $1$ and $0$ .

$lim_{x \to 2^{+}} \frac{\frac{x}{{(x^{2} - 4)}^{\frac{1}{2}}}}{1}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Rewrite ${(x^{2} - 4)}^{\frac{1}{2}}$ as $\sqrt{x^{2} - 4}$ .

$lim_{x \to 2^{+}} \frac{x}{\sqrt{x^{2} - 4}} \cdot 1$

Step 1.6

Multiply $\frac{x}{\sqrt{x^{2} - 4}}$ by $1$ .

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

Step 2

Since the numerator is positive and the denominator $\sqrt{x^{2} - 4}$ approaches zero and is greater than zero for $x$ near $2$ to the right, the function increases without bound.

$\infty$