Question

The value of $\underset{x\to 0}{lim}\frac{1+\sin x-\cos x+\log (1-x)}{x^3}$

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $0$
• D. None of these

Answer 1

The correct option is B $-\frac{1}{2}$

$\underset{x\to 0}{lim}\frac{1+\sin x-\cos x+\log (1-x)}{x^3}$
$=\underset{x\to 0}{lim}\frac{1+(x-\frac{x^3}{3!}+\dots )-(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots )+(-x-\frac{x^2}{2}-\frac{x^3}{3}-\dots )}{x^3}$
$=-\frac{1}{3!}-\frac{1}{3}=-\frac{1}{2}$