Question

Evaluate the limit: $\underset{x\to 2}{lim}\frac{x^4-16}{x-2}$

Answer 1

Given: $\underset{x\to 2}{lim}\frac{x^4-16}{x-2}becomes\frac{0}{0}form$

Using Factorization method

$\Rightarrow \underset{x\to 2}{lim}\frac{(x^2{)}^2-4^2}{x-2}[\because (a^2-b^2)=(a+b\left)\right(a-b\left)\right]$

$\Rightarrow \underset{x\to 2}{lim}\frac{(x^2-4)(x^2+4)}{x-2}$

$\Rightarrow \underset{x\to 2}{lim}\frac{(x+2)(x-2)(x^2+4)}{x-2}\{x\ne 2\}$

$\Rightarrow \underset{x\to 2}{lim}(x+2)(x^2+4)=(2+2)(4+4)$

$=4\left(8\right)=32$

Therefore , $\underset{x\to 2}{lim}\frac{x^4-16}{x-2}=32$