# Find the limit. lim_{xrightarrow(pi/2)}frac{sec x}{tan x}

Question
Limits and continuity
Find the limit.
$$\lim_{x\rightarrow(\pi/2)}\frac{\sec x}{\tan x}$$

2021-01-28
We have to evaluate
$$\lim_{x\rightarrow(\pi/2)^-}\frac{\sec x}{\tan x}$$
We can see that if we put given limit then we get $$\frac{\infty}{\infty}$$ form.
So, to remove this form, we will simplify the given expression and then we will put limit.
We know that $$\sec x=\frac{1}{\cos x}$$ and $$\tan x=\frac{\sin x}{\cos x}$$
Therefore,
$$\lim_{x\rightarrow(\pi/2)^-}\frac{\sec x}{\tan x}=\lim_{x\rightarrow(\pi/2)^-}\frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}}$$
$$\lim_{x\rightarrow\pi/2}\frac{1}{\sin x}$$
$$=\frac{1}{\sin{\frac{\pi}{2}}}$$
$$=\frac{1}{1}$$
$$=1$$
Hence, required answer is 1 .

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