Differentiation Question 468
Question: $ \underset{x\to 0}{\mathop{\lim }},\frac{\sin x^{4}-x^{4}\cos x^{4}+x^{20}}{x^{4}({e^{2x^{4}}}1-2x^{4})} $ is equal to
Options:
A) 0
B) $ -1/6 $
C) $ 1/6 $
D) Does not exist
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ \underset{x\to 0}{\mathop{\lim }},\frac{\sin x^{4}-x^{4}\cos x^{4}+x^{20}}{x^{4}({e^{2x^{4}}}1-2x^{4})} $ $ =\underset{t\to 0}{\mathop{\lim }},\frac{\sin t-t\cos t+t^{5}}{t(e^{2t}-1-2t)} $ $ =\underset{t\to 0}{\mathop{\lim }},\frac{t-\frac{t^{3}}{3!}+\frac{t^{5}}{5!}…-t( 1-\frac{t^{2}}{2!}+\frac{t^{4}}{4!}… )+t^{5}}{t( 1+2t+\frac{4t^{2}}{2!}+\frac{8t^{3}}{3!}+\frac{16t^{4}}{4!}+…-1-2t )} $ $ =\underset{t\to 0}{\mathop{\lim }},\frac{-\frac{t^{3}}{6}+\frac{t^{3}}{2}+\frac{t^{5}}{5!}-\frac{t^{5}}{4!}+…+t^{5}}{2t^{3}+\frac{8t^{4}}{3!}+….} $ $ =\frac{-\frac{1}{6}+\frac{1}{2}}{2}=-\frac{-1+3}{12}=\frac{1}{6} $
BETA
