Calculus Examples

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

Step 1

Apply trigonometric identities.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

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

$lim_{x \to 0} \sec (x)$

Step 2

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

$\sec (lim_{x \to 0} x)$

Step 3

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

$\sec (0)$

Step 4

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

$1$