Calculus Examples

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

Step 1

Apply trigonometric identities.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

$lim_{x \to \frac{π}{2}} \frac{\cos (x)}{1} \cdot \frac{\sin (x)}{\cos (x)}$

Step 1.4

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

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Step 1.4.1

Cancel the common factor.

$lim_{x \to \frac{π}{2}} \frac{\cos (x)}{1} \cdot \frac{\sin (x)}{\cos (x)}$

Step 1.4.2

Rewrite the expression.

$lim_{x \to \frac{π}{2}} \sin (x)$

Step 2

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

$\sin (lim_{x \to \frac{π}{2}} x)$

Step 3

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\sin (\frac{π}{2})$

Step 4

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$1$