Question

Evaluate the limit as x turns to 2⁺

lim {sin[In (x-1)] / √(x-2)}

Answer 1

$\lim\limits_{x\rightarrow2^+} \frac{sin(ln(x-1))}{\sqrt{x-2}}\\ =\lim\limits_{x\rightarrow2^+} \frac{sin(ln(x-1))}{(ln(x-1))} \times \frac{(ln(x-1))}{\sqrt{x-2}}\\ =\lim\limits_{x\rightarrow2^+} \frac{sin(ln(x-1))}{(ln(x-1))} \times \lim\limits_{x\rightarrow2^+}\frac{(ln(x-1))}{\sqrt{x-2}}\\ =(1)\times \lim\limits_{x\rightarrow2^+}\frac{(ln(x-1))}{\sqrt{x-2}}\\$

Now, applying L'Hopital's rule, we get:

$\lim\limits_{x\rightarrow2^+} \frac{\frac{1}{x-1}}{\frac{1}{2\sqrt{x-2}}}\\ =\lim\limits_{x\rightarrow2^+} \frac{2\sqrt{x-2}}{x-1}\\ =\frac{2(\sqrt{2-2})}{2-1}\\ =0$