First slide

Introduction to limits

Question

$lim_{x \to \infty} \{\frac{x^{4} \sin (\frac{1}{x}) + x^{2}}{1 + | x |^{3}}\}$ is

Moderate

A

2
B

1
C

-1
D

does not exist

Solution

$lim_{x \to \infty} [\frac{x^{4} \sin (\frac{1}{x}) + x^{2}}{(1 + | x |^{3})}]$
$Let L = lim_{x \to - \infty} \frac{x^{3}}{1 + | x |^{3}} [xsin (\frac{1}{x}) + \frac{1}{x}]$
$= lim_{x \to - \infty} \frac{x^{3}}{| x |^{3}} [\frac{1}{1 + \frac{1}{| x | x}}] [xsin (\frac{1}{x}) + \frac{1}{x}]$
now,
$\begin{array}{l} lim_{x \to - \infty} \frac{x^{3}}{| x |^{3}} = lim_{x \to - \infty} \frac{x^{3}}{- x^{3}} = - 1 [∵ | x | = - x for x < 0] \\ lim_{x \to - \infty} \frac{1}{| x | x} = 0 \end{array}$
and $lim_{x \to - \infty} xsin (\frac{1}{x}) = lim_{y \to 0^{-}} \frac{\sin y}{y} = 1 [where y = \frac{1}{x}]$
From eq (i) $L = - 1 (\frac{1}{1 + 0}) (1 + 0) = - 1$

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