Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

The exact value of $\sin (0)$ is $0$ .

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

Step 5.1.2

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

$\frac{0 - 1 \cdot 1}{x}$

Step 5.1.3

Multiply $- 1$ by $1$ .

$\frac{0 - 1}{x}$

Step 5.1.4

Subtract $1$ from $0$ .

$\frac{- 1}{x}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{1}{x}$