Chapter 1: Q1P (page 1)

In the following problems, find the limit of the given sequence as $n \to \infty$ .
1. $\frac{n^{2} + 5 n^{3}}{2 n^{3} + 3 \sqrt{4 + n^{6}}}$

Short Answer

Expert verified

The limit of the given sequence $\frac{n^{2} + 5 n^{3}}{2 n^{3} + 3 \sqrt{4 + n^{6}}}$ as $n \to \infty$ is 1.

Step by step solution

Definition of limits

A limit of a function/expression/sequence is used when the value of the function/sequence cannot be calculated directly or is difficult to calculate at a particular value. It is generally defined as the output obtained when the input is very close to the input value.

Given parameters

The given expression is $\frac{n^{2} + 5 n^{3}}{2 n^{3} + 3 \sqrt{4 + n^{6}}}$ and the limit is $n \to \infty$ .

Solve the limits

Divide the numerator and the denominator of the given expression by $n^{3}$ .

$\frac{\frac{n^{2} + 5 n^{3}}{n^{3}}}{\frac{2 n^{3} + 3 \sqrt{4 + n^{6}}}{n^{3}}} = \frac{\frac{1}{n} + 5}{2 + \frac{3}{n^{3}} \sqrt{4 + n^{6}}}$

Take $\frac{1}{n^{3}}$ inside the radical.

$\begin{array}{rcl} \frac{\frac{1}{n} + 5}{2 + \frac{3}{n^{3}} \sqrt{4 + n^{6}}} & = & \frac{\frac{1}{n} + 5}{2 + 3 \sqrt{\frac{4}{n^{6}} + \frac{n^{6}}{n^{6}}}} \\ = & \frac{\frac{1}{n} + 5}{2 + 3 \sqrt{\frac{4}{n^{6}} + 1}} \end{array}$

Apply the limit $n \to \infty$ in the obtained expression.

$\lim_{n \to \infty} \frac{\frac{1}{n} + 5}{2 + 3 \sqrt{\frac{4}{n^{6}} + 1}}$

Substitute 0 for $\frac{1}{n}$ into the obtained expression and solve.

$\begin{array}{rcl} \lim_{n \to \infty} \frac{\frac{1}{n} + 5}{2 + 3 \sqrt{\frac{4}{n^{6}} + 1}} & = & \frac{0 + 5}{2 + 3 \sqrt{0 + 1}} \\ = & \frac{5}{2 + 3} \\ = & 1 \end{array}$

Thus, the limit of the given expression is 1.

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In the following problems, find the limit of the given sequence as $n \to \infty$ .
1. $\frac{n^{2} + 5 n^{3}}{2 n^{3} + 3 \sqrt{4 + n^{6}}}$

Short Answer

Step by step solution

Definition of limits

Given parameters

Solve the limits

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In the following problems, find the limit of the given sequence as n→∞.1.n2+5n32n3+34+n6

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Step by step solution

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Given parameters

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In the following problems, find the limit of the given sequence as $n \to \infty$ .
1. $\frac{n^{2} + 5 n^{3}}{2 n^{3} + 3 \sqrt{4 + n^{6}}}$