Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{x - \tan (x)}$

Step 4.1.3

Add $0$ and $0$ .

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

Step 4.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 4.3

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

$0$