Calculus Examples

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

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 $0$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{1 - 1 \cdot 1}{\sin (x)}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{\sin (x)}$

Step 3.1.3

Subtract $1$ from $1$ .

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

Step 3.2

Separate fractions.

$\frac{0}{1} \cdot \frac{1}{\sin (x)}$

Step 3.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{0}{1} \csc (x)$

Step 3.4

Divide $0$ by $1$ .

$0 \csc (x)$

Step 3.5

Multiply $0$ by $\csc (x)$ .

$0$