Calculus Examples

$lim_{h \to 0} \frac{\ln (2 + h) - \ln (2)}{h}$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$lim_{h \to 0} \frac{\ln (\frac{2 + h}{2})}{h}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{lim_{h \to 0} \ln (\frac{2 + h}{2})}{lim_{h \to 0} h}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Move the limit inside the logarithm.

$\frac{\ln (lim_{h \to 0} \frac{2 + h}{2})}{lim_{h \to 0} h}$

Step 2.1.2.1.2

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

$\frac{\ln (\frac{1}{2} lim_{h \to 0} 2 + h)}{lim_{h \to 0} h}$

Step 2.1.2.1.3

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

$\frac{\ln (\frac{1}{2} (lim_{h \to 0} 2 + lim_{h \to 0} h))}{lim_{h \to 0} h}$

Step 2.1.2.1.4

Evaluate the limit of $2$ which is constant as $h$ approaches $0$ .

$\frac{\ln (\frac{1}{2} (2 + lim_{h \to 0} h))}{lim_{h \to 0} h}$

Step 2.1.2.2

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

$\frac{\ln (\frac{1}{2} (2 + 0))}{lim_{h \to 0} h}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Add $2$ and $0$ .

$\frac{\ln (\frac{1}{2} \cdot 2)}{lim_{h \to 0} h}$

Step 2.1.2.3.2

Cancel the common factor of $2$ .

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Step 2.1.2.3.2.1

Cancel the common factor.

$\frac{\ln (\frac{1}{2} \cdot 2)}{lim_{h \to 0} h}$

Step 2.1.2.3.2.2

Rewrite the expression.

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

Step 2.1.2.3.3

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

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

Step 2.1.3

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

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

$lim_{h \to 0} \frac{\ln (\frac{2 + h}{2})}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [\ln (\frac{2 + h}{2})]}{\frac{d}{d h} [h]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [\ln (\frac{2 + h}{2})]}{\frac{d}{d h} [h]}$

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = \ln (h)$ and $g (h) = \frac{2 + h}{2}$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $\frac{2 + h}{2}$ .

$lim_{h \to 0} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d h} [\frac{2 + h}{2}]}{\frac{d}{d h} [h]}$

Step 2.3.2.2

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

$lim_{h \to 0} \frac{\frac{1}{u} \frac{d}{d h} [\frac{2 + h}{2}]}{\frac{d}{d h} [h]}$

Step 2.3.2.3

Replace all occurrences of $u$ with $\frac{2 + h}{2}$ .

$lim_{h \to 0} \frac{\frac{1}{\frac{2 + h}{2}} \frac{d}{d h} [\frac{2 + h}{2}]}{\frac{d}{d h} [h]}$

Step 2.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{2 + h}{2}$ .

$lim_{h \to 0} \frac{1 \frac{2}{2 + h} \frac{d}{d h} [\frac{2 + h}{2}]}{\frac{d}{d h} [h]}$

Step 2.3.4

Multiply $\frac{2}{2 + h}$ by $1$ .

$lim_{h \to 0} \frac{\frac{2}{2 + h} \frac{d}{d h} [\frac{2 + h}{2}]}{\frac{d}{d h} [h]}$

Step 2.3.5

Since $\frac{1}{2}$ is constant with respect to $h$ , the derivative of $\frac{2 + h}{2}$ with respect to $h$ is $\frac{1}{2} \frac{d}{d h} [2 + h]$ .

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

Step 2.3.6

Multiply $\frac{1}{2}$ by $\frac{2}{2 + h}$ .

$lim_{h \to 0} \frac{\frac{2}{2 (2 + h)} \frac{d}{d h} [2 + h]}{\frac{d}{d h} [h]}$

Step 2.3.7

Cancel the common factor of $2$ .

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Step 2.3.7.1

Cancel the common factor.

$lim_{h \to 0} \frac{\frac{2}{2 (2 + h)} \frac{d}{d h} [2 + h]}{\frac{d}{d h} [h]}$

Step 2.3.7.2

Rewrite the expression.

$lim_{h \to 0} \frac{\frac{1}{2 + h} \frac{d}{d h} [2 + h]}{\frac{d}{d h} [h]}$

Step 2.3.8

By the Sum Rule, the derivative of $2 + h$ with respect to $h$ is $\frac{d}{d h} [2] + \frac{d}{d h} [h]$ .

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

Step 2.3.9

Since $2$ is constant with respect to $h$ , the derivative of $2$ with respect to $h$ is $0$ .

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

Step 2.3.10

Add $0$ and $\frac{d}{d h} [h]$ .

$lim_{h \to 0} \frac{\frac{1}{2 + h} \frac{d}{d h} [h]}{\frac{d}{d h} [h]}$

Step 2.3.11

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

$lim_{h \to 0} \frac{\frac{1}{2 + h} \cdot 1}{\frac{d}{d h} [h]}$

Step 2.3.12

Multiply $\frac{1}{2 + h}$ by $1$ .

$lim_{h \to 0} \frac{\frac{1}{2 + h}}{\frac{d}{d h} [h]}$

Step 2.3.13

Reorder terms.

$lim_{h \to 0} \frac{\frac{1}{h + 2}}{\frac{d}{d h} [h]}$

Step 2.3.14

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

$lim_{h \to 0} \frac{\frac{1}{h + 2}}{1}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} \frac{1}{h + 2} \cdot 1$

Step 2.5

Multiply $\frac{1}{h + 2}$ by $1$ .

$lim_{h \to 0} \frac{1}{h + 2}$

Step 3

Evaluate the limit.

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Step 3.1

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

$\frac{lim_{h \to 0} 1}{lim_{h \to 0} h + 2}$

Step 3.2

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

$\frac{1}{lim_{h \to 0} h + 2}$

Step 3.3

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

$\frac{1}{lim_{h \to 0} h + lim_{h \to 0} 2}$

Step 3.4

Evaluate the limit of $2$ which is constant as $h$ approaches $0$ .

$\frac{1}{lim_{h \to 0} h + 2}$

Step 4

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

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

Step 5

Add $0$ and $2$ .

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$