Question

Evaluate $\lim _{x \rightarrow 1}\left(\frac{y-4 \sqrt{y}+3}{y^{2}-1}\right)$

Answer 1

$\lim _{x \rightarrow 1}\left(\frac{y-4 \sqrt{y}+3}{y^{2}-1}\right)\left(y-4 y^{1 / 2}+3\right.)$
$\lim _{x \rightarrow 1\left[\frac{\left(y^{k / 2}-3\right)\left(y^{1 / 2}-1\right)}{(y-1)(y+1)}\right]}$
$\therefore \lim _{x \rightarrow 1}\left[\frac{\left(y^{1 / 2}-3\right)\left(y^{1 / 2}-1\right)}{ \left.\left(y^{1 / 2}+1\right)\left(y^{1 / 2}-1\right)(y+1)\right]}\right]$
$\therefore \quad \lim _{x \rightarrow 1}\left[\frac{y^{1 / 2}-3}{\left(y^{1 / 2}+1\right)(y+1)}\right]$
$\quad \frac{1^{1 / 2}-3}{\left(1^{1 / 2}+1\right)(1+1)}=\frac{-2}{(2)(2)}$

$=\dfrac{1}{2}$