First slide

Introduction to limits

Question

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

Moderate

A

0
B

-1
C

2
D

1

Solution

$lim_{x \to \infty} \frac{x^{4} \cdot \sin (\frac{1}{x}) + x^{2}}{1 + | x |^{3}} = lim_{x \to \infty} [\frac{xsin (\frac{1}{x}) + \frac{1}{x}}{\frac{1}{x^{3}} + \frac{| x |^{3}}{x^{3}}}]$
[dividing numerator and denominator by y²]
$\begin{array}{l} = \frac{lim_{x \to \infty} \frac{\sin (\frac{1}{x})}{\frac{1}{x}} + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \frac{1}{x^{3}} + lim_{x \to \infty} \frac{| x |^{3}}{x^{3}}} \\ = \frac{1 - 0}{0 + 1} = 1 \end{array}$

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