_ x 1 1+ x ^2 x is equal to - Vidyadip

MCQ

$\lim _{x \rightarrow 1} \frac{1+\cos \pi x}{\tan ^2 \pi x}$ is equal to

A
$0$
✓
$\frac{1}{2}$
C
1
D
2

✓

Answer

Correct option: B.

$\frac{1}{2}$

(B)
Applying L-Hospital's rule, we get
$\lim _{x \rightarrow 1} \frac{1+\cos \pi x}{\tan ^2 \pi x}=\lim _{x \rightarrow 1} \frac{-\pi \sin \pi x}{2 \pi \tan \pi x \sec ^2 \pi x}$
$=\frac{-1}{2} \lim _{x \rightarrow 1} \cos ^3 \pi x$
$=-\frac{1}{2}(-1)^3=\frac{1}{2}$

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