Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(1 + \arctan (x))}^{\frac{1}{x}}$ as $e^{\ln ({(1 + \arctan (x))}^{\frac{1}{x}})}$ .

$lim_{x \to 0} e^{\ln ({(1 + \arctan (x))}^{\frac{1}{x}})}$

Step 1.2

Expand $\ln ({(1 + \arctan (x))}^{\frac{1}{x}})$ by moving $\frac{1}{x}$ outside the logarithm.

$lim_{x \to 0} e^{\frac{1}{x} \ln (1 + \arctan (x))}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 0} \frac{1}{x} \ln (1 + \arctan (x))}$

Step 2.2

Combine $\frac{1}{x}$ and $\ln (1 + \arctan (x))$ .

$e^{lim_{x \to 0} \frac{\ln (1 + \arctan (x))}{x}}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

$e^{\frac{lim_{x \to 0} \ln (1 + \arctan (x))}{lim_{x \to 0} x}}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Move the limit inside the logarithm.

$e^{\frac{\ln (lim_{x \to 0} 1 + \arctan (x))}{lim_{x \to 0} x}}$

Step 3.1.2.2

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

$e^{\frac{\ln (lim_{x \to 0} 1 + lim_{x \to 0} \arctan (x))}{lim_{x \to 0} x}}$

Step 3.1.2.3

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

$e^{\frac{\ln (1 + lim_{x \to 0} \arctan (x))}{lim_{x \to 0} x}}$

Step 3.1.2.4

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

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Step 3.1.2.4.1

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

$e^{\frac{\ln (1 + \arctan (0))}{lim_{x \to 0} x}}$

Step 3.1.2.4.2

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

$e^{\frac{\ln (1 + 0)}{lim_{x \to 0} x}}$

Step 3.1.2.5

Simplify the answer.

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Step 3.1.2.5.1

Add $1$ and $0$ .

$e^{\frac{\ln (1)}{lim_{x \to 0} x}}$

Step 3.1.2.5.2

The natural logarithm of $1$ is $0$ .

$e^{\frac{0}{lim_{x \to 0} x}}$

Step 3.1.3

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

$e^{\frac{0}{0}}$

Step 3.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{\frac{0}{0}}$

Step 3.2

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.

$lim_{x \to 0} \frac{\ln (1 + \arctan (x))}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (1 + \arctan (x))]}{\frac{d}{d x} [x]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$e^{lim_{x \to 0} \frac{\frac{d}{d x} [\ln (1 + \arctan (x))]}{\frac{d}{d x} [x]}}$

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 $f (x) = \ln (x)$ and $g (x) = 1 + \arctan (x)$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $1 + \arctan (x)$ .

$e^{lim_{x \to 0} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [1 + \arctan (x)]}{\frac{d}{d x} [x]}}$

Step 3.3.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 3.3.2.3

Replace all occurrences of $u$ with $1 + \arctan (x)$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Add $0$ and $\frac{d}{d x} [\arctan (x)]$ .

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

Step 3.3.6

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

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

Step 3.3.7

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

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

Step 3.3.8

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

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

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

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

Add $1$ and $0$ .

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

Step 6.1.2

Multiply $1 + 0^{2}$ by $1$ .

$e^{\frac{1}{1 + 0^{2}}}$

Step 6.1.3

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

$e^{\frac{1}{1 + 0}}$

Step 6.1.4

Add $1$ and $0$ .

$e^{\frac{1}{1}}$

Step 6.2

Divide $1$ by $1$ .

$e^{1}$

Step 6.3

Simplify.

$e$

Step 7

The result can be shown in multiple forms.

Exact Form:

$e$

Decimal Form:

$2.71828182 \dots$