Question

Estimating limits numerically and graphically.

Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically.

$\underset{x\to \infty }{\lim }\left(\sqrt{9x^2+x}-3x\right)$

Short answer

The value of limit of the given function is $\infty .$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{x\to \infty }{\lim }\left(\sqrt{9x^2+x}-3x\right)$

Step 2. Formula used.

Here, we simplify the fractions.

Step 3. Simplification.

Given,

$\begin{array}{l}\underset{x\to \infty }{\lim }\sqrt{9x^2+x}-3x\\ =\underset{x\to \infty }{\lim }\left\{\sqrt{9x^2\left(1+\frac{x}{9x^2}\right)}-3x\right\}\\ =\underset{x\to \infty }{\lim }\left\{\sqrt{9x^2}\sqrt{1+\frac{1}{9x}}-3x\right\}\\ =\underset{x\to \infty }{\lim }\left\{3x\sqrt{1+\frac{1}{9x}}-3x\right\}\\ =\infty \end{array}$

Therefore, our final answer is $\infty .$