Question

If $f\left(x\right)=\frac{sin\left(e^{x-2}\right)}{log(x-1)},$ the $\underset{x\to 2}{L}tf\left(x\right)$ is given by

Answer 1

$f\left(x\right)=\frac{sin\left(e^{x-2}\right)}{cos(x-1)}Apply{L}^{\prime }Hosprctorrule\underset{x\to 2}{lt}\frac{cos\left(e^{\frac{x-2}{-1}}\right)}{\left(\frac{1}{x-1}\right)}\frac{cos(e^{2-2}-1).e^{2-2}}{\frac{1}{2.1}}=\frac{cos0.e^{\circ }}{1}=1$