Question

If $f\left(1\right)=1,f^{\prime }\left(1\right)=2$, then write the value of $li{m}_{x\to 1}\frac{\sqrt{f\left(x\right)}-1}{\sqrt{x}-1}$

Answer 1

Given

$f\left(1\right)=1$

$f^{\prime }\left(1\right)=2$

$\text{Now,}$

$li{m}_{x\to 1}\frac{\sqrt{f\left(x\right)}-1}{\sqrt{x}-1}=li{m}_{x\to 1}\frac{\frac{f^{\prime }\left(x\right)}{2\sqrt{f\left(x\right)}}}{\frac{1}{2\sqrt{x}}}$

$=li{m}_{x\to 1}\frac{f^{\prime }\left(x\right)\sqrt{x}}{\sqrt{f\left(x\right)}}=\frac{f^{\prime }\left(1\right)\sqrt{1}}{\sqrt{f\left(1\right)}}=2$