Calculus Examples

$lim_{x \to 0} \frac{2 \sin (x)}{\cos (x) \sin (x)}$

Step 1

Evaluate the limit.

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Step 1.1

Cancel the common factor of $\sin (x)$ .

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Step 1.1.1

Cancel the common factor.

$lim_{x \to 0} \frac{2 \sin (x)}{\cos (x) \sin (x)}$

Step 1.1.2

Rewrite the expression.

$lim_{x \to 0} \frac{2}{\cos (x)}$

Step 1.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 0} \frac{1}{\cos (x)}$

Step 1.3

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

$2 \frac{lim_{x \to 0} 1}{lim_{x \to 0} \cos (x)}$

Step 1.4

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

$2 \frac{1}{lim_{x \to 0} \cos (x)}$

Step 1.5

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

$2 \frac{1}{\cos (lim_{x \to 0} x)}$

Step 2

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

$2 \frac{1}{\cos (0)}$

Step 3

Simplify the answer.

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Step 3.1

Convert from $\frac{1}{\cos (0)}$ to $\sec (0)$ .

$2 \sec (0)$

Step 3.2

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

$2 \cdot 1$

Step 3.3

Multiply $2$ by $1$ .

$2$