Calculus Examples

$lim_{x \to 0} \frac{\arctan (x)}{\sin (x)}$

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.

$\frac{lim_{x \to 0} \arctan (x)}{lim_{x \to 0} \sin (x)}$

Step 1.1.2

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

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Step 1.1.2.1

Evaluate the limit of $\arctan (x)$ by plugging in $0$ for $x$ .

$\frac{\arctan (0)}{lim_{x \to 0} \sin (x)}$

Step 1.1.2.2

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

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\arctan (x)}{\sin (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\arctan (x)]}{\frac{d}{d x} [\sin (x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\arctan (x)]}{\frac{d}{d x} [\sin (x)]}$

Step 1.3.2

The derivative of $\arctan (x)$ with respect to $x$ is $\frac{1}{1 + x^{2}}$ .

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

Step 1.3.3

Reorder terms.

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

Step 1.3.4

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

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

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

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 $0$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

$\frac{1}{(0^{2} + 1) \cdot \cos (0)}$

Step 4

Simplify the answer.

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Step 4.1

Separate fractions.

$\frac{1}{0^{2} + 1} \cdot \frac{1}{\cos (0)}$

Step 4.2

Convert from $\frac{1}{\cos (0)}$ to $\sec (0)$ .

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

Step 4.3

Simplify the denominator.

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Step 4.3.1

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

$\frac{1}{0 + 1} \sec (0)$

Step 4.3.2

Add $0$ and $1$ .

$\frac{1}{1} \sec (0)$

Step 4.4

Cancel the common factor of $1$ .

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Step 4.4.1

Cancel the common factor.

$\frac{1}{1} \sec (0)$

Step 4.4.2

Rewrite the expression.

$1 \sec (0)$

Step 4.5

Multiply $\sec (0)$ by $1$ .

$\sec (0)$

Step 4.6

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

$1$