Question

Evaluate;-
$\underset{x\to 0}{lim}{\left(\cos x+\sin x\right)}^{\frac{1}{x}}$

• A. $-e$
• B. $e$
• C. $2e$
• D. $3e$

Answer 1

Answer: (B)

$e$

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

this is a limit of form $1^{\infty }$

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

$lnL=\underset{x\to 0}{lim}\frac{\cos x+\sin x-1}{x}$

Now this is of form $\%$

$lnL=\underset{x\to 0}{lim}\frac{-\sin x+\cos x}{1}$

$=1$

$L=e^1=e$