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

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

Question
Limits and continuity
asked 2021-01-27
Find the limit.
\(\lim_{x\rightarrow(\pi/2)}\frac{\sec x}{\tan x}\)

Answers (1)

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 .
0

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