find-lim-x-0-sinx-cosx-1-x-

Question Number 30502 by abdo imad last updated on 22/Feb/18

f i n d {l i m}_{x \to 0} {(s i n x + c o s x)}^{\frac{1}{x}} .

Answered by sma3l2996 last updated on 24/Feb/18

{(s i n x + c o s x)}^{1 / x} = e^{\frac{l n (s i n x + c o s x)}{x}}

s i n (x) \underset{0}{\sim} x - \frac{x^{3}}{3!} a n d c o s (x) \underset{0}{\sim} 1 - \frac{x^{2}}{2}

s o s i n x + c o s x \underset{0}{\sim} 1 + x - \frac{x^{2}}{2} - \frac{x^{3}}{6}

l n (c o s x + s i n x) \underset{0}{\sim} l n (1 + x - x^{2} / 2 - x^{3} / 6) \underset{0}{\sim} x - x^{2} / 2 - x^{3} / 6

\frac{l n (s i n x + c o s x)}{x} \underset{0}{\sim} 1

s o \underset{x \to 0}{l i m} {(s i n x + c o s x)}^{1 / x} = e