Calculus Examples

$lim_{x \to 0} \frac{x^{2}}{1 - \sqrt{1 + x^{2}}}$

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.

$\frac{lim_{x \to 0} x^{2}}{lim_{x \to 0} 1 - \sqrt{1 + x^{2}}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$\frac{0^{2}}{lim_{x \to 0} 1 - \sqrt{1 + x^{2}}}$

Step 1.1.2.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{lim_{x \to 0} 1 - \sqrt{1 + x^{2}}}$

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 $0$ .

$\frac{0}{lim_{x \to 0} 1 - lim_{x \to 0} \sqrt{1 + x^{2}}}$

Step 1.1.3.1.2

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

$\frac{0}{1 - lim_{x \to 0} \sqrt{1 + x^{2}}}$

Step 1.1.3.1.3

Move the limit under the radical sign.

$\frac{0}{1 - \sqrt{lim_{x \to 0} 1 + x^{2}}}$

Step 1.1.3.1.4

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

$\frac{0}{1 - \sqrt{lim_{x \to 0} 1 + lim_{x \to 0} x^{2}}}$

Step 1.1.3.1.5

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

$\frac{0}{1 - \sqrt{1 + lim_{x \to 0} x^{2}}}$

Step 1.1.3.1.6

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

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

Step 1.1.3.2

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

$\frac{0}{1 - \sqrt{1 + 0^{2}}}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{1 - \sqrt{1 + 0}}$

Step 1.1.3.3.1.2

Add $1$ and $0$ .

$\frac{0}{1 - \sqrt{1}}$

Step 1.1.3.3.1.3

Any root of $1$ is $1$ .

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

Step 1.1.3.3.1.4

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{x^{2}}{1 - \sqrt{1 + x^{2}}} = lim_{x \to 0} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [1 - \sqrt{1 + x^{2}}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

By the Sum Rule, the derivative of $1 - \sqrt{1 + x^{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt{1 + x^{2}}]$ .

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

Step 1.3.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.3.5

Evaluate $\frac{d}{d x} [- \sqrt{1 + x^{2}}]$ .

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Step 1.3.5.1

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

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

Step 1.3.5.2

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

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

Step 1.3.5.3

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 $g (x) = 1 + x^{2}$ .

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Step 1.3.5.3.1

To apply the Chain Rule, set $u$ as $1 + x^{2}$ .

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

Step 1.3.5.3.2

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

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

Step 1.3.5.3.3

Replace all occurrences of $u$ with $1 + x^{2}$ .

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

Step 1.3.5.4

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$ .

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

Step 1.3.5.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.3.5.6

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

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

Step 1.3.5.7

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

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

Step 1.3.5.8

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

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

Step 1.3.5.9

Combine the numerators over the common denominator.

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

Step 1.3.5.10

Simplify the numerator.

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Step 1.3.5.10.1

Multiply $- 1$ by $2$ .

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

Step 1.3.5.10.2

Subtract $2$ from $1$ .

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

Step 1.3.5.11

Move the negative in front of the fraction.

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

Step 1.3.5.12

Add $0$ and $2 x$ .

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

Step 1.3.5.13

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

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

Step 1.3.5.14

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

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

Step 1.3.5.15

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

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

Step 1.3.5.16

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

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

Step 1.3.5.17

Cancel the common factor.

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

Step 1.3.5.18

Rewrite the expression.

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

Step 1.3.6

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

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

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

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

Step 1.6

Combine factors.

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Step 1.6.1

Multiply $- 1$ by $2$ .

$lim_{x \to 0} - 2 x \frac{\sqrt{1 + x^{2}}}{x}$

Step 1.6.2

Combine $- 2$ and $\frac{\sqrt{1 + x^{2}}}{x}$ .

$lim_{x \to 0} x \frac{- 2 \sqrt{1 + x^{2}}}{x}$

Step 1.6.3

Combine $x$ and $\frac{- 2 \sqrt{1 + x^{2}}}{x}$ .

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

Step 1.7

Cancel the common factor of $x$ .

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Step 1.7.1

Cancel the common factor.

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

Step 1.7.2

Divide $- 2 \sqrt{1 + x^{2}}$ by $1$ .

$lim_{x \to 0} - 2 \sqrt{1 + x^{2}}$

Step 2

Evaluate the limit.

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Step 2.1

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

$- 2 lim_{x \to 0} \sqrt{1 + x^{2}}$

Step 2.2

Move the limit under the radical sign.

$- 2 \sqrt{lim_{x \to 0} 1 + x^{2}}$

Step 2.3

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

$- 2 \sqrt{lim_{x \to 0} 1 + lim_{x \to 0} x^{2}}$

Step 2.4

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

$- 2 \sqrt{1 + lim_{x \to 0} x^{2}}$

Step 2.5

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

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

Step 3

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

$- 2 \sqrt{1 + 0^{2}}$

Step 4

Simplify the answer.

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Step 4.1

Raising $0$ to any positive power yields $0$ .

$- 2 \sqrt{1 + 0}$

Step 4.2

Add $1$ and $0$ .

$- 2 \sqrt{1}$

Step 4.3

Any root of $1$ is $1$ .

$- 2 \cdot 1$

Step 4.4

Multiply $- 2$ by $1$ .

$- 2$