Question

In the following problems, find the limit of the given sequence as $\mathit{n}\mathbf{\to }\mathbf{\infty }$.

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

Short answer

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

Solve the limits

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

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

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.

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

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

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

Thus, the limit of the given expression is 1.