Calculus Examples

$\frac{lim_{x \to 0} 3 x - \sin (3 x)}{3 x - \tan (3 x)}$

Step 1

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

$\frac{lim_{x \to 0} 3 x - lim_{x \to 0} \sin (3 x)}{3 x - \tan (3 x)}$

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 0} x - lim_{x \to 0} \sin (3 x)}{3 x - \tan (3 x)}$

Step 3

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

$\frac{3 lim_{x \to 0} x - \sin (lim_{x \to 0} 3 x)}{3 x - \tan (3 x)}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 0} x - \sin (3 lim_{x \to 0} x)}{3 x - \tan (3 x)}$

Step 5

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

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Step 5.1

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

$\frac{3 \cdot 0 - \sin (3 lim_{x \to 0} x)}{3 x - \tan (3 x)}$

Step 5.2

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

$\frac{3 \cdot 0 - \sin (3 \cdot 0)}{3 x - \tan (3 x)}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $3$ by $0$ .

$\frac{0 - \sin (3 \cdot 0)}{3 x - \tan (3 x)}$

Step 6.1.2

Multiply $3$ by $0$ .

$\frac{0 - \sin (0)}{3 x - \tan (3 x)}$

Step 6.1.3

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

$\frac{0 - 0}{3 x - \tan (3 x)}$

Step 6.1.4

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{3 x - \tan (3 x)}$

Step 6.1.5

Add $0$ and $0$ .

$\frac{0}{3 x - \tan (3 x)}$

Step 6.2

Rewrite $\tan (3 x)$ in terms of sines and cosines.

$\frac{0}{3 x - \frac{\sin (3 x)}{\cos (3 x)}}$

Step 6.3

Divide $0$ by $3 x - \frac{\sin (3 x)}{\cos (3 x)}$ .

$0$