Calculus Examples

$\frac{lim_{x \to 0} 5 - \cos (x)}{x}$

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 $x$ approaches $0$ .

$\frac{lim_{x \to 0} 5 - lim_{x \to 0} \cos (x)}{x}$

Step 1.2

Evaluate the limit of $5$ which is constant as $x$ approaches $0$ .

$\frac{5 - lim_{x \to 0} \cos (x)}{x}$

Step 1.3

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

$\frac{5 - \cos (lim_{x \to 0} x)}{x}$

Step 2

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

$\frac{5 - \cos (0)}{x}$

Step 3

Simplify the numerator.

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Step 3.1

The exact value of $\cos (0)$ is $1$ .

$\frac{5 - 1 \cdot 1}{x}$

Step 3.2

Multiply $- 1$ by $1$ .

$\frac{5 - 1}{x}$

Step 3.3

Subtract $1$ from $5$ .

$\frac{4}{x}$