Question

find the limit.

$\underset{x\to \infty }{\lim }\frac{2x+1}{5x-1}$

Short answer

The value of limit of the given function is$\frac{2}{5}.$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{x\to \infty }{\lim }\frac{2x+1}{5x-1}$

Step 2. Formula used.

Here, in the given question we apply limit of a quotient and limit of sum,difference.

After it we apply the formula:

$\underset{x\to \infty }{\lim }\frac{1}{x}=0$

Step 3. Simplification.

Given, $\underset{x\to \infty }{\lim }\frac{2x+1}{5x-1}$

Here, first we divide the numerator and denominator by the greatest power of given in the question then we apply limit of constant multiple.

After it we apply the value of $x$which is equals to $\infty .$

So, divide the numerator and denominator by $x$.

$\begin{array}{c}\underset{x\to \infty }{\lim }\frac{2x+1}{5x-1}=\underset{x\to \infty }{\lim }\left(\frac{2+\frac{1}{x}}{5-\frac{1}{x}}\right)\\ =\left(\frac{\underset{x\to \infty }{\lim }2+\underset{x\to \infty }{\lim }\frac{1}{x}}{\underset{x\to \infty }{\lim }5-\underset{x\to \infty }{\lim }\frac{1}{x}}\right)\\ =\frac{2+0}{5-0}\\ =\frac{2}{5}\end{array}$

Therefore, our final answer is$\frac{2}{5}.$