Calculus Examples

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

Step 1

Apply trigonometric identities.

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Step 1.1

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

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

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 1.4

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$lim_{x \to 0} \frac{\cot (x)}{\csc (x)}$

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Step 1.8

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 1.9

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

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Step 1.9.1

Cancel the common factor.

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

Step 1.9.2

Rewrite the expression.

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

Step 2

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

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

Step 3

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

$\cos (0)$

Step 4

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

$1$