Question

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

• A. $-e$
• B. $e^{-2}$
• C. $e$
• D. $1/e$

Answer 1

The correct option is C $e$

Since, the function is of the form, $1^{\infty }$

We rewrite the expression in the form,

${lim}_{x\to 0}exp\left[\frac{sinx+cosx-1}{x}\right]$

Since, the limit now is of the form, $\frac{0}{0}$

We apply L-Hospital's rule, to get,

${lim}_{x\to 0}exp[cosx-sinx]$

Now applying the limit, we get the answer as

$=e$