Functions Question 138
Question: $ \underset{x\to 0}{\mathop{\lim }},\frac{{e^{x^{2}}}-\cos x}{x^{2}}= $
[IIT Screening]
Options:
A) $ \frac{3}{2} $
B) $ -\frac{1}{2} $
C) 1
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
$ x=\frac{1}{t}, $ Now expanding $ {e^{x^{2}}} $ and $ \cos x, $ we get $ \underset{x\to 0}{\mathop{\lim }}\frac{\frac{3x^{2}}{2,!}+x^{4},( \frac{1}{2,!}-\frac{1}{4,!} )+…….}{x^{2}}=\frac{3}{2} $ Aliter : Apply L-Hospital?s rule, $ \underset{x\to 0}{\mathop{\lim }}\frac{2x{e^{x^{2}}}+\sin x}{2x}=\underset{x\to 0}{\mathop{\lim }}{e^{x^{2}}}+\underset{x\to 0}{\mathop{\lim }}\frac{\sin x}{2x}=1+\frac{1}{2}=\frac{3}{2}. $
BETA
