Calculus Examples

$\frac{lim_{x \to 0} \cos (5 x) - 1}{2 x}$

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 $0$ .

$\frac{lim_{x \to 0} \cos (5 x) - lim_{x \to 0} 1}{2 x}$

Step 1.2

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

$\frac{\cos (lim_{x \to 0} 5 x) - lim_{x \to 0} 1}{2 x}$

Step 1.3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{\cos (5 lim_{x \to 0} x) - lim_{x \to 0} 1}{2 x}$

Step 1.4

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

$\frac{\cos (5 lim_{x \to 0} x) - 1 \cdot 1}{2 x}$

Step 2

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

$\frac{\cos (5 \cdot 0) - 1 \cdot 1}{2 x}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $5$ by $0$ .

$\frac{\cos (0) - 1 \cdot 1}{2 x}$

Step 3.1.2

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

$\frac{1 - 1 \cdot 1}{2 x}$

Step 3.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{2 x}$

Step 3.1.4

Subtract $1$ from $1$ .

$\frac{0}{2 x}$

Step 3.2

Cancel the common factor of $0$ and $2$ .

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Step 3.2.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 x}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $2$ out of $2 x$ .

$\frac{2 (0)}{2 (x)}$

Step 3.2.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 x}$

Step 3.2.2.3

Rewrite the expression.

$\frac{0}{x}$

Step 3.3

Divide $0$ by $x$ .

$0$