Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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 $x$ .

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

Step 4.2

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

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

Step 5

Simplify the answer.

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Step 5.1

Separate fractions.

$\frac{3 \cdot 0}{1} \cdot \frac{\tan (0)}{\sin (x)}$

Step 5.2

Rewrite $\tan (0)$ in terms of sines and cosines.

$\frac{3 \cdot 0}{1} \cdot \frac{\frac{\sin (0)}{\cos (0)}}{\sin (x)}$

Step 5.3

Rewrite $\frac{\frac{\sin (0)}{\cos (0)}}{\sin (x)}$ as a product.

$\frac{3 \cdot 0}{1} (\frac{\sin (0)}{\cos (0)} \cdot \frac{1}{\sin (x)})$

Step 5.4

Multiply $\frac{\sin (0)}{\cos (0)}$ by $\frac{1}{\sin (x)}$ .

$\frac{3 \cdot 0}{1} \cdot \frac{\sin (0)}{\cos (0) \sin (x)}$

Step 5.5

Multiply $3$ by $0$ .

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

Step 5.6

Divide $0$ by $1$ .

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

Step 5.7

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

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

Step 5.8

Simplify the denominator.

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Step 5.8.1

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

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

Step 5.8.2

Multiply $\sin (x)$ by $1$ .

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

Step 5.9

Divide $0$ by $\sin (x)$ .

$0 \cdot 0$

Step 5.10

Multiply $0$ by $0$ .

$0$