First slide

Introduction to limits

Question

$lim_{x \to \infty} \frac{\sqrt{x^{2} + 1} - \sqrt[3]{x^{3} + 1}}{\sqrt[4]{x^{4} + 1} - \sqrt[5]{x^{4} + 1}}$ equals

Moderate

A

1
B

0
C

-1
D

none of these

Solution

We have,
$\begin{array}{l} lim_{x \to \infty} \frac{\sqrt{x^{2} + 1} - \sqrt[3]{x^{3} + 1}}{\sqrt[4]{x^{4} + 1} - \sqrt[5]{x^{4} + 1}} \\ = lim_{x \to \infty} \frac{\sqrt{1 + \frac{1}{x^{2}}} - \sqrt[3]{1 + \frac{1}{x^{3}}}}{\sqrt[4]{1 + \frac{1}{x^{4}} - \sqrt[5]{\frac{1}{x} + \frac{1}{x^{5}}}}} = \frac{1 - 1}{1 - 0} = 0 \end{array}$

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