Calculus Examples

$lim_{x \to 2} \frac{\sqrt{x^{2} + 21} - 5}{x - 2}$

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.

$\frac{lim_{x \to 2} \sqrt{x^{2} + 21} - 5}{lim_{x \to 2} x - 2}$

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 $2$ .

$\frac{lim_{x \to 2} \sqrt{x^{2} + 21} - lim_{x \to 2} 5}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 2} x^{2} + 21} - lim_{x \to 2} 5}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.3

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

$\frac{\sqrt{lim_{x \to 2} x^{2} + lim_{x \to 2} 21} - lim_{x \to 2} 5}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.4

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

$\frac{\sqrt{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 21} - lim_{x \to 2} 5}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.5

Evaluate the limit of $21$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{{(lim_{x \to 2} x)}^{2} + 21} - lim_{x \to 2} 5}{lim_{x \to 2} x - 2}$

Step 1.1.2.1.6

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

$\frac{\sqrt{{(lim_{x \to 2} x)}^{2} + 21} - 1 \cdot 5}{lim_{x \to 2} x - 2}$

Step 1.1.2.2

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

$\frac{\sqrt{2^{2} + 21} - 1 \cdot 5}{lim_{x \to 2} x - 2}$

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

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

$\frac{\sqrt{4 + 21} - 1 \cdot 5}{lim_{x \to 2} x - 2}$

Step 1.1.2.3.1.2

Add $4$ and $21$ .

$\frac{\sqrt{25} - 1 \cdot 5}{lim_{x \to 2} x - 2}$

Step 1.1.2.3.1.3

Rewrite $25$ as $5^{2}$ .

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

Step 1.1.2.3.1.4

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

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

Step 1.1.2.3.1.5

Multiply $- 1$ by $5$ .

$\frac{5 - 5}{lim_{x \to 2} x - 2}$

Step 1.1.2.3.2

Subtract $5$ from $5$ .

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

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.

$lim_{x \to 2} \frac{\sqrt{x^{2} + 21} - 5}{x - 2} = lim_{x \to 2} \frac{\frac{d}{d x} [\sqrt{x^{2} + 21} - 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 2} \frac{\frac{d}{d x} [\sqrt{x^{2} + 21} - 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt{x^{2} + 21} - 5$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x^{2} + 21}] + \frac{d}{d x} [- 5]$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [\sqrt{x^{2} + 21}] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [\sqrt{x^{2} + 21}]$ .

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Step 1.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 21}$ as ${(x^{2} + 21)}^{\frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [{(x^{2} + 21)}^{\frac{1}{2}}] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.2

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 $g (x) = x^{2} + 21$ .

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Step 1.3.3.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 21$ .

$lim_{x \to 2} \frac{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} + 21] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.2.2

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

$lim_{x \to 2} \frac{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 21] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.2.3

Replace all occurrences of $u$ with $x^{2} + 21$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 21] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.3

By the Sum Rule, the derivative of $x^{2} + 21$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [21]$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [21]) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.4

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

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{\frac{1}{2} - 1} (2 x + \frac{d}{d x} [21]) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.5

Since $21$ is constant with respect to $x$ , the derivative of $21$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{\frac{1}{2} - 1} (2 x + 0) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.6

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

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (2 x + 0) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.7

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

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (2 x + 0) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.8

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{\frac{1 - 1 \cdot 2}{2}} (2 x + 0) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.9

Simplify the numerator.

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Step 1.3.3.9.1

Multiply $- 1$ by $2$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{\frac{1 - 2}{2}} (2 x + 0) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.9.2

Subtract $2$ from $1$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{\frac{- 1}{2}} (2 x + 0) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.10

Move the negative in front of the fraction.

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{- \frac{1}{2}} (2 x + 0) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.11

Add $2 x$ and $0$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(x^{2} + 21)}^{- \frac{1}{2}} (2 x) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.12

Combine $\frac{1}{2}$ and ${(x^{2} + 21)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{{(x^{2} + 21)}^{- \frac{1}{2}}}{2} (2 x) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.13

Combine $2$ and $\frac{{(x^{2} + 21)}^{- \frac{1}{2}}}{2}$ .

$lim_{x \to 2} \frac{\frac{2 {(x^{2} + 21)}^{- \frac{1}{2}}}{2} x + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.14

Combine $\frac{2 {(x^{2} + 21)}^{- \frac{1}{2}}}{2}$ and $x$ .

$lim_{x \to 2} \frac{\frac{2 {(x^{2} + 21)}^{- \frac{1}{2}} x}{2} + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.15

Move ${(x^{2} + 21)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 2} \frac{\frac{2 x}{2 {(x^{2} + 21)}^{\frac{1}{2}}} + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.16

Cancel the common factor.

$lim_{x \to 2} \frac{\frac{2 x}{2 {(x^{2} + 21)}^{\frac{1}{2}}} + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.17

Rewrite the expression.

$lim_{x \to 2} \frac{\frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}} + \frac{d}{d x} [- 5]}{\frac{d}{d x} [x - 2]}$

Step 1.3.4

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

$lim_{x \to 2} \frac{\frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}} + 0}{\frac{d}{d x} [x - 2]}$

Step 1.3.5

Add $\frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 2} \frac{\frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}}}{\frac{d}{d x} [x - 2]}$

Step 1.3.6

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

$lim_{x \to 2} \frac{\frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 2]}$

Step 1.3.7

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

$lim_{x \to 2} \frac{\frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}}}{1 + \frac{d}{d x} [- 2]}$

Step 1.3.8

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

$lim_{x \to 2} \frac{\frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}}}{1 + 0}$

Step 1.3.9

Add $1$ and $0$ .

$lim_{x \to 2} \frac{\frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}}}{1}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 2} \frac{x}{{(x^{2} + 21)}^{\frac{1}{2}}} \cdot 1$

Step 1.5

Rewrite ${(x^{2} + 21)}^{\frac{1}{2}}$ as $\sqrt{x^{2} + 21}$ .

$lim_{x \to 2} \frac{x}{\sqrt{x^{2} + 21}} \cdot 1$

Step 1.6

Multiply $\frac{x}{\sqrt{x^{2} + 21}}$ by $1$ .

$lim_{x \to 2} \frac{x}{\sqrt{x^{2} + 21}}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{lim_{x \to 2} x}{lim_{x \to 2} \sqrt{x^{2} + 21}}$

Step 2.2

Move the limit under the radical sign.

$\frac{lim_{x \to 2} x}{\sqrt{lim_{x \to 2} x^{2} + 21}}$

Step 2.3

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

$\frac{lim_{x \to 2} x}{\sqrt{lim_{x \to 2} x^{2} + lim_{x \to 2} 21}}$

Step 2.4

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

$\frac{lim_{x \to 2} x}{\sqrt{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 21}}$

Step 2.5

Evaluate the limit of $21$ which is constant as $x$ approaches $2$ .

$\frac{lim_{x \to 2} x}{\sqrt{{(lim_{x \to 2} x)}^{2} + 21}}$

Step 3

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

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Step 3.1

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

$\frac{2}{\sqrt{{(lim_{x \to 2} x)}^{2} + 21}}$

Step 3.2

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

$\frac{2}{\sqrt{2^{2} + 21}}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{2}{\sqrt{4 + 21}}$

Step 4.2

Add $4$ and $21$ .

$\frac{2}{\sqrt{25}}$

Step 4.3

Rewrite $25$ as $5^{2}$ .

$\frac{2}{\sqrt{5^{2}}}$

Step 4.4

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

$\frac{2}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{5}$

Decimal Form:

$0.4$