Q1P In the following problems, find ... [FREE SOLUTION]

Chapter 1: Q1P (page 1)

In the following problems, find the limit of the given sequence as $n 鈫� 鈭�$ .
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 $n 鈫� 鈭�$ 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 $n 鈫� 鈭�$ .

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 $n 鈫� 鈭�$ in the obtained expression.

$\lim_{n 鈫� 鈭�} \frac{\frac{1}{n} + 5}{2 + 3 \sqrt{\frac{4}{n^{6}} + 1}}$

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

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

Thus, the limit of the given expression is 1.

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91影视

In the following problems, find the limit of the given sequence as $n 鈫� 鈭�$ .
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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91影视

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

Short Answer

Step by step solution

Definition of limits

Given parameters

Solve the limits

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Quantum Physics

Electricity and Magnetism

Electric Charge Field and Potential

Further Mechanics and Thermal Physics

Geometrical and Physical Optics

Measurements

Study anywhere. Anytime. Across all devices.

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