First slide

Introduction to limits

Question

The value of $lim_{x \to \infty} \frac{\sqrt{n^{2} + 1} + \sqrt{n}}{\sqrt[4]{n^{4} + n} + \sqrt[4]{n}}$ , is

Moderate

A

0
B

1
C

-1
D

none of these

Solution

we have,
$\begin{array}{l} lim_{n \to \infty} \frac{\sqrt{n^{2} + 1} + \sqrt{n}}{\sqrt[4]{n^{3} + n - \sqrt[4]{n}}} \\ = lim_{n \to \infty} \frac{n \{\sqrt{1 + \frac{1}{n^{2}}} + \frac{1}{\sqrt{n}}\}}{n^{3 / 4} \{\sqrt[4]{1 + \frac{1}{n^{2}}} - \frac{1}{\sqrt{n}}\}} \\ = lim_{n \to \infty} n^{1 / 4} \frac{\{\sqrt{1 + \frac{1}{n^{2}}} + \frac{1}{\sqrt{n}}\}}{\{\sqrt[4]{1 + \frac{1}{n^{2}}} - \frac{1}{\sqrt{n}}\}} = \infty \times 2 = \infty \end{array}$

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