Calculus Examples

$lim_{x \to \frac{π}{2}} \frac{\cos (x)}{\sin (22 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\frac{\cos (x)}{\sin (22 x)}$ as a product.

$lim_{x \to \frac{π}{2}} \cos (x) \frac{1}{\sin (22 x)}$

Step 1.2

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

$lim_{x \to \frac{π}{2}} \frac{\cos (x)}{1} \cdot \frac{1}{\sin (22 x)}$

Step 1.3

Simplify.

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Step 1.3.1

Divide $\cos (x)$ by $1$ .

$lim_{x \to \frac{π}{2}} \cos (x) \frac{1}{\sin (22 x)}$

Step 1.3.2

Convert from $\frac{1}{\sin (22 x)}$ to $\csc (22 x)$ .

$lim_{x \to \frac{π}{2}} \cos (x) \csc (22 x)$

Step 2

Consider the left sided limit.

$lim_{x \to {(\frac{π}{2})}^{-}} \cos (x) \csc (22 x)$

Step 3

Make a table to show the behavior of the function $\cos (x) \csc (22 x)$ as $x$ approaches $\frac{π}{2}$ from the left.

$\begin{array}{c} x & \cos (x) \csc (22 x) \\ 1.47079632 & 0.12348034 \\ 1.56079632 & 0.04582252 \\ 1.56979632 & 0.0454582 \\ 1.57069632 & 0.04545458 \end{array}$

Step 4

As the $x$ values approach $\frac{π}{2}$ , the function values approach $0.045$ . Thus, the limit of $\cos (x) \csc (22 x)$ as $x$ approaches $\frac{π}{2}$ from the left is $0.045$ .

$0.045$

Step 5

Consider the right sided limit.

$lim_{x \to {(\frac{π}{2})}^{+}} \cos (x) \csc (22 x)$

Step 6

Make a table to show the behavior of the function $\cos (x) \csc (22 x)$ as $x$ approaches $\frac{π}{2}$ from the right.

$\begin{array}{c} x & \cos (x) \csc (22 x) \\ 1.67079632 & 0.12348034 \\ 1.58079632 & 0.04582252 \\ 1.57179632 & 0.0454582 \\ 1.57089632 & 0.04545458 \end{array}$

Step 7

As the $x$ values approach $\frac{π}{2}$ , the function values approach $0.045$ . Thus, the limit of $\cos (x) \csc (22 x)$ as $x$ approaches $\frac{π}{2}$ from the right is $0.045$ .

$0.045$