Question

Evaluate the limit: $\underset{x\to \frac{1}{4}}{lim}\frac{4x-1}{2\sqrt{x}-1}$

Answer 1

Given: $\underset{x\to \frac{1}{4}}{lim}\frac{4x-1}{2\sqrt{x}-1}$ becomes $\frac{0}{0}$ form

Using Factorization method

$\underset{x\to \frac{1}{4}}{lim}\frac{4x-1}{2\sqrt{x}-1}=\underset{x\to \frac{1}{4}}{lim}\frac{{\left(2\sqrt{x}\right)}^2-1^2}{2\sqrt{x}-1}$

$\underset{x\to \frac{1}{4}}{lim}\frac{(2\sqrt{x}-1)(2\sqrt{x}+1)}{2\sqrt{x}-1}$

$[\because (a^2-b^2)=(a+b\left)\right(a-b\left)\right]$

$\underset{x\to \frac{1}{4}}{lim}(2\sqrt{x}+1)$

$=(2\sqrt{\frac{1}{4}}+1)=(2\left(\frac{1}{2}\right)+1)$

$=1+1=2$

$Therefore,$

$\underset{x\to \frac{1}{4}}{lim}\frac{4x-1}{2\sqrt{x}-1}=2$