Limits Question 9
Question 9 - 2024 (31 Jan Shift 1)
$\lim _{x \rightarrow 0} \frac{e^{4 \sin x \mid}-2|\sin x|-1}{x^{2}}$
(1) is equal to -1
(2) does not exist
(3) is equal to 1
(4) is equal to 2
Show Answer
Answer (4)
Solution
$\lim _{x \rightarrow 0} \frac{e^{\tan x \mid}-2|\sin x|-1}{x^{2}}$
$\lim _{x \rightarrow 0} \frac{e^{4 \sin x}-2|\sin x|-1}{|\sin x|^{2}} \times \frac{\sin ^{2} x}{x^{2}}$
Let $|\sin \mathrm{x}|=\mathrm{t}$
$\lim _{t \rightarrow 0} \frac{e^{2 t}-2 t-1}{t^{2}} \times \lim _{x \rightarrow 0} \frac{\sin ^{2} x}{x^{2}}$
$=\lim _{t \rightarrow 0} \frac{2 e^{2 t}-2}{2 t} \times 1=2 \times 1=2$
