Calculus Examples

$lim_{x \to 0} \frac{3 x}{\ln (x + 1)}$

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.

$\frac{lim_{x \to 0} 3 x}{lim_{x \to 0} \ln (x + 1)}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 0} x}{lim_{x \to 0} \ln (x + 1)}$

Step 1.2.2

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

$\frac{3 \cdot 0}{lim_{x \to 0} \ln (x + 1)}$

Step 1.2.3

Multiply $3$ by $0$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Move the limit inside the logarithm.

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.2

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

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

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Add $0$ and $1$ .

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

Step 1.3.3.2

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

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{3 x}{\ln (x + 1)} = lim_{x \to 0} \frac{\frac{d}{d x} [3 x]}{\frac{d}{d x} [\ln (x + 1)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [3 x]}{\frac{d}{d x} [\ln (x + 1)]}$

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{3 \frac{d}{d x} [x]}{\frac{d}{d x} [\ln (x + 1)]}$

Step 3.3

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

$lim_{x \to 0} \frac{3 \cdot 1}{\frac{d}{d x} [\ln (x + 1)]}$

Step 3.4

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{3}{\frac{d}{d x} [\ln (x + 1)]}$

Step 3.5

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 3.5.1

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

$lim_{x \to 0} \frac{3}{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + 1]}$

Step 3.5.2

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

$lim_{x \to 0} \frac{3}{\frac{1}{u} \frac{d}{d x} [x + 1]}$

Step 3.5.3

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

$lim_{x \to 0} \frac{3}{\frac{1}{x + 1} \frac{d}{d x} [x + 1]}$

Step 3.6

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

$lim_{x \to 0} \frac{3}{\frac{1}{x + 1} (\frac{d}{d x} [x] + \frac{d}{d x} [1])}$

Step 3.7

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

$lim_{x \to 0} \frac{3}{\frac{1}{x + 1} (1 + \frac{d}{d x} [1])}$

Step 3.8

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

$lim_{x \to 0} \frac{3}{\frac{1}{x + 1} (1 + 0)}$

Step 3.9

Add $1$ and $0$ .

$lim_{x \to 0} \frac{3}{\frac{1}{x + 1} \cdot 1}$

Step 3.10

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

$lim_{x \to 0} \frac{3}{\frac{1}{x + 1}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} 3 (x + 1)$

Step 5

Evaluate the limit.

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Step 5.1

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 0} x + 1$

Step 5.2

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

$3 (lim_{x \to 0} x + lim_{x \to 0} 1)$

Step 5.3

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

$3 (lim_{x \to 0} x + 1)$

Step 6

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

$3 (0 + 1)$

Step 7

Simplify the answer.

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Step 7.1

Add $0$ and $1$ .

$3 \cdot 1$

Step 7.2

Multiply $3$ by $1$ .

$3$