limx→−∞{x4sin(1x)+x21+|x|3} is equal to

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Question

lim x \to - \infty {\frac{x^{4} sin (\frac{1}{x}) + x^{2}}{1 + | x |^{3}}}

is equal to

Options:

A . 2

B . 1

C . −1

D . does not exist

Answer: Option C
:
C

lim x \to - \infty [\frac{x^{4} sin (\frac{1}{x}) + x^{2}}{1 + | x |^{3}}]

lim x \to - \infty \frac{x^{3}}{1 + | x |^{3}} [x sin (\frac{1}{x}) + \frac{1}{x}]

As

x \to - \infty

\frac{x^{3}}{1 + | x |^{3}} = - 1

x s i n (\frac{1}{x}) + \frac{1}{x} = 1

\Rightarrow l i m x \to - \infty \frac{x^{3}}{1 + | x |^{3}} (x s i n \frac{1}{x} + \frac{1}{x}) = - 1

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