Question

Evaluate:

$\underset{x\to \frac{\pi }{2}}{lim}\tan x=$

• A. $\infty$
• B. $-\infty$
• C. 0
• D. does not exist

Answer 1

The correct option is D does not exist

$\underset{x\to \frac{\pi }{2}}{lim}\tan x$

$RHL=\underset{h\to 0^{+}}{lim}\tan (\frac{\pi }{2}+h)$

$=\underset{h\to 0^{+}}{lim}-\cot h$

$=\underset{h\to 0^{+}}{lim}\frac{-\cot h}{h}\times h=-1\times 0=0$

$LHL=\underset{h\to 0^{-}}{lim}\tan (\frac{\pi }{2}-h)$

$=\underset{h\to 0^{-}}{lim}\cot h$

$=\underset{h\to 0^{-}}{lim}\frac{\cot h}{h}\times h=1\times 0=0$

Thus,$LHL=RHL$

Left hand limit = right hand limit $=f\left(a\right)$

if the above condition is satisfied then limit exists.

$f\left(\frac{\pi }{2}\right)=\underset{x\to \frac{\pi }{2}}{lim}\tan x=\tan \frac{\pi }{2}=\infty$

Left hand limit = right hand limit $\ne f\left(a\right)$

Hence the limit does not exists.