Calculus Examples

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

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.

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

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

Move the limit inside the logarithm.

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Add $0$ and $1$ .

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

Step 1.2.3.2

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

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

Step 1.3

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

$\frac{0}{0}$

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 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 $g (x) = x + 1$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

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

Step 3.2.2

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

$lim_{x \to 0} \frac{\frac{1}{u} \frac{d}{d x} [x + 1]}{\frac{d}{d x} [x]}$

Step 3.2.3

Replace all occurrences of $u$ with $x + 1$ .

$lim_{x \to 0} \frac{\frac{1}{x + 1} \frac{d}{d x} [x + 1]}{\frac{d}{d x} [x]}$

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $0$ .

$lim_{x \to 0} \frac{\frac{1}{x + 1} \cdot 1}{\frac{d}{d x} [x]}$

Step 3.7

Multiply $\frac{1}{x + 1}$ by $1$ .

$lim_{x \to 0} \frac{\frac{1}{x + 1}}{\frac{d}{d x} [x]}$

Step 3.8

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

$lim_{x \to 0} \frac{\frac{1}{x + 1}}{1}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{1}{x + 1} \cdot 1$

Step 5

Evaluate the limit.

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Step 5.1

Multiply $\frac{1}{x + 1}$ by $1$ .

$lim_{x \to 0} \frac{1}{x + 1}$

Step 5.2

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

$\frac{lim_{x \to 0} 1}{lim_{x \to 0} x + 1}$

Step 5.3

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

$\frac{1}{lim_{x \to 0} x + 1}$

Step 5.4

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

$\frac{1}{lim_{x \to 0} x + lim_{x \to 0} 1}$

Step 5.5

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

$\frac{1}{lim_{x \to 0} x + 1}$

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

Add $0$ and $1$ .

$\frac{1}{1}$

Step 7.2

Divide $1$ by $1$ .

$1$