Question

Solve $\underset{x\to 1}{lim}\frac{1+\cos \pi x}{(1-x)2}$

Answer 1

$\underset{x\to 1}{lim}\frac{1+\cos \pi x}{{\left(1-x\right)}^2}$ $\left(÷form\right)$

using L- hospital rule

$=\underset{x\to 1}{lim}\frac{-\sin \pi x\times \pi }{2\left(1-x\right)\times -1}$

$=\underset{x\to 1}{lim}\frac{\pi \sin \pi x}{2\left(1-x\right)}(÷)$

again using L-Hospital rule

$\underset{x\to 1}{lim}\frac{{\pi }^2\cos \pi x}{2\left(-1\right)}$

$=\frac{{\pi }^2-1}{-2}$

$=\frac{{\pi }^2}{2}$