Mathematics - Limits Question with Solution | TestHub

Given, $\underset{x\to 1}{\lim }\frac{\left(x^2-1\right){\sin }^2\left(\pi x\right)}{x^4-2x^3+2x-1}$

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

$=\underset{h\to 0}{\lim }\frac{{\sin }^2\left(\pi \left(1+h\right)\right)}{{\left(1+h-1\right)}^2}$

$=\underset{h\to 0}{\lim }\frac{{\sin }^2\pi h}{h^2}=\underset{h\to 0}{\lim }{\left(\frac{\sin \pi h}{\pi h}\right)}^2{\pi }^2={\pi }^2$