Question

Evaluate:

$\underset{x\to 0}{lim}\frac{x-\log (1+x)}{1-\cos x}$

Answer 1

Given,

$\underset{x\to 0}{lim}\frac{x-\log (1+x)}{1-\cos x}$ $\left(\frac{0}{0}\right)$ form.

Now applying L'Hospital's rule we get,

$=\underset{x\to 0}{lim}\frac{1-\frac{1}{(1+x)}}{\sin x}$ $\left(\frac{0}{0}\right)$ form.

Now applying L'Hospital's rule we get,

$=\underset{x\to 0}{lim}\frac{\frac{1}{(1+x{)}^2}}{\cos x}$

$=1$.