Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

Move the limit inside the logarithm.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Evaluate the limit of $\ln (\cos (x))$ which is constant as $h$ approaches $0$ .

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

Step 2

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

$\frac{\ln (\cos (x + 0)) - \ln (\cos (x))}{h}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $x$ and $0$ .

$\frac{\ln (\cos (x)) - \ln (\cos (x))}{h}$

Step 3.1.2

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

$\frac{\ln (\frac{\cos (x)}{\cos (x)})}{h}$

Step 3.1.3

Cancel the common factor of $\cos (x)$ .

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Step 3.1.3.1

Cancel the common factor.

$\frac{\ln (\frac{\cos (x)}{\cos (x)})}{h}$

Step 3.1.3.2

Rewrite the expression.

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

Step 3.1.4

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

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$