Calculus Examples

$lim_{x \to 0} \frac{\sqrt{x + 121} - 11}{\tan (x)}$

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.

$\frac{lim_{x \to 0} \sqrt{x + 121} - 11}{lim_{x \to 0} \tan (x)}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

$\frac{lim_{x \to 0} \sqrt{x + 121} - lim_{x \to 0} 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} x + 121} - lim_{x \to 0} 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.1.3

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

$\frac{\sqrt{lim_{x \to 0} x + lim_{x \to 0} 121} - lim_{x \to 0} 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.1.4

Evaluate the limit of $121$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + 121} - lim_{x \to 0} 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.1.5

Evaluate the limit of $11$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + 121} - 1 \cdot 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.2

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

$\frac{\sqrt{0 + 121} - 1 \cdot 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Add $0$ and $121$ .

$\frac{\sqrt{121} - 1 \cdot 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.3.1.2

Rewrite $121$ as $11^{2}$ .

$\frac{\sqrt{11^{2}} - 1 \cdot 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.3.1.3

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

$\frac{11 - 1 \cdot 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.3.1.4

Multiply $- 1$ by $11$ .

$\frac{11 - 11}{lim_{x \to 0} \tan (x)}$

Step 1.2.3.2

Subtract $11$ from $11$ .

$\frac{0}{lim_{x \to 0} \tan (x)}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Move the limit inside the trig function because tangent is continuous.

$\frac{0}{\tan (lim_{x \to 0} x)}$

Step 1.3.2

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

$\frac{0}{\tan (0)}$

Step 1.3.3

The exact value of $\tan (0)$ is $0$ .

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\sqrt{x + 121} - 11}{\tan (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x + 121} - 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x + 121} - 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.2

By the Sum Rule, the derivative of $\sqrt{x + 121} - 11$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x + 121}] + \frac{d}{d x} [- 11]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x + 121}] + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3

Evaluate $\frac{d}{d x} [\sqrt{x + 121}]$ .

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 121}$ as ${(x + 121)}^{\frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [{(x + 121)}^{\frac{1}{2}}] + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

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 $g (x) = x + 121$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $x + 121$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x + 121] + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.2.2

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

$lim_{x \to 0} \frac{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x + 121] + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.2.3

Replace all occurrences of $u$ with $x + 121$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 121] + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.3

By the Sum Rule, the derivative of $x + 121$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [121]$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [121]) + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.4

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

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} [121]) + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.5

Since $121$ is constant with respect to $x$ , the derivative of $121$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{\frac{1}{2} - 1} (1 + 0) + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.6

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

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0) + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.7

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

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.8

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.9

Simplify the numerator.

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Step 3.3.9.1

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{\frac{1 - 2}{2}} (1 + 0) + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.9.2

Subtract $2$ from $1$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{\frac{- 1}{2}} (1 + 0) + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.10

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{- \frac{1}{2}} (1 + 0) + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.11

Add $1$ and $0$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 121)}^{- \frac{1}{2}} \cdot 1 + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.12

Combine $\frac{1}{2}$ and ${(x + 121)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{{(x + 121)}^{- \frac{1}{2}}}{2} \cdot 1 + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.13

Multiply $\frac{{(x + 121)}^{- \frac{1}{2}}}{2}$ by $1$ .

$lim_{x \to 0} \frac{\frac{{(x + 121)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.3.14

Move ${(x + 121)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 121)}^{\frac{1}{2}}} + \frac{d}{d x} [- 11]}{\frac{d}{d x} [\tan (x)]}$

Step 3.4

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 121)}^{\frac{1}{2}}} + 0}{\frac{d}{d x} [\tan (x)]}$

Step 3.5

Add $\frac{1}{2 {(x + 121)}^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 121)}^{\frac{1}{2}}}}{\frac{d}{d x} [\tan (x)]}$

Step 3.6

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 121)}^{\frac{1}{2}}}}{\sec^{2} (x)}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{1}{2 {(x + 121)}^{\frac{1}{2}}} \cdot \frac{1}{\sec^{2} (x)}$

Step 5

Rewrite ${(x + 121)}^{\frac{1}{2}}$ as $\sqrt{x + 121}$ .

$lim_{x \to 0} \frac{1}{2 \sqrt{x + 121}} \cdot \frac{1}{\sec^{2} (x)}$

Step 6

Multiply $\frac{1}{2 \sqrt{x + 121}}$ by $\frac{1}{\sec^{2} (x)}$ .

$lim_{x \to 0} \frac{1}{2 \sqrt{x + 121} \sec^{2} (x)}$

Step 7

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 0} \frac{1}{\sqrt{x + 121} \sec^{2} (x)}$

Step 8

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

$\frac{1}{2} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} \sqrt{x + 121} \sec^{2} (x)}$

Step 9

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

$\frac{1}{2} \cdot \frac{1}{lim_{x \to 0} \sqrt{x + 121} \sec^{2} (x)}$

Step 10

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

$\frac{1}{2} \cdot \frac{1}{lim_{x \to 0} \sqrt{x + 121} \cdot lim_{x \to 0} \sec^{2} (x)}$

Step 11

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} x + 121} \cdot lim_{x \to 0} \sec^{2} (x)}$

Step 12

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

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} x + lim_{x \to 0} 121} \cdot lim_{x \to 0} \sec^{2} (x)}$

Step 13

Evaluate the limit of $121$ which is constant as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} x + 121} \cdot lim_{x \to 0} \sec^{2} (x)}$

Step 14

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

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} x + 121} \cdot {(lim_{x \to 0} \sec (x))}^{2}}$

Step 15

Move the limit inside the trig function because secant is continuous.

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 0} x + 121} \cdot \sec^{2} (lim_{x \to 0} x)}$

Step 16

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

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Step 16.1

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

$\frac{1}{2} \cdot \frac{1}{\sqrt{0 + 121} \cdot \sec^{2} (lim_{x \to 0} x)}$

Step 16.2

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

$\frac{1}{2} \cdot \frac{1}{\sqrt{0 + 121} \cdot \sec^{2} (0)}$

Step 17

Simplify the answer.

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Step 17.1

Combine.

$\frac{1 \cdot 1}{2 (\sqrt{0 + 121} \cdot \sec^{2} (0))}$

Step 17.2

Multiply $1$ by $1$ .

$\frac{1}{2 \sqrt{0 + 121} \cdot \sec^{2} (0)}$

Step 17.3

Simplify the denominator.

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Step 17.3.1

Add $0$ and $121$ .

$\frac{1}{2 \sqrt{121} \sec^{2} (0)}$

Step 17.3.2

Rewrite $121$ as $11^{2}$ .

$\frac{1}{2 \sqrt{11^{2}} \sec^{2} (0)}$

Step 17.3.3

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

$\frac{1}{2 \cdot 11 \sec^{2} (0)}$

Step 17.3.4

The exact value of $\sec (0)$ is $1$ .

$\frac{1}{2 \cdot 11 \cdot 1^{2}}$

Step 17.3.5

One to any power is one.

$\frac{1}{2 \cdot 11 \cdot 1}$

Step 17.3.6

Combine exponents.

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Step 17.3.6.1

Multiply $2$ by $11$ .

$\frac{1}{22 \cdot 1}$

Step 17.3.6.2

Multiply $22$ by $1$ .

$\frac{1}{22}$