Question

$\underset{x\to 0}{\lim }\frac{sin{x}^4-x^{4}cos{x}^4+x^{20}}{x^4\left(e^{2x^{4}1-2x^4}\right)}$ is equal to

• A. 0
• B. - 1/6
• C. 1/6
• D. does not exist

Answer 1

Answer: (C)

$\underset{x\to 0}{\lim }\frac{sin{x}^4-x^{4}cos{x}^4+x^{20}}{x^4\left(e^{2x^{4}1-2x^4}\right)}$
$=\underset{t\to 0}{\lim }\frac{sint-tcost+t^5}{t\left(e^{2t}-1-2t\right)}$
$=\underset{t\to 0}{\lim }\frac{t-\frac{t^3}{3!}+\frac{t^5}{5!}.....-t\left(1-\frac{t^2}{2!}+\frac{t^4}{4!}..........\right)+t^5}{t\left(1+2t+\frac{4t^2}{2!}+\frac{8t^3}{3!}+\frac{16t^4}{4!}+....-1-2t\right)}$
$=\underset{t\to 0}{\lim }\frac{-\frac{t^3}{6}+\frac{t^3}{2}+\frac{t^5}{5!}-\frac{t^5}{4!}+.....+t^5}{2t^3+\frac{8t^4}{3!}+.......}$
$=\frac{-\frac{1}{6}+\frac{1}{2}}{2}=-\frac{-1+3}{12}=\frac{1}{6}$