Question

use the properties of limits to help decide whether the limit exists. if the limit exists, find its value.
limₓ→∞ 4x/(5x - 2)
select the correct choice below and, if necessary, fill in the answer box within your choice.
a. limₓ→∞ 4x/(5x - 2) = (simplify your answer. type an integer or a fraction.)
b. the limit does not exist and is neither ∞ nor -∞.

Explanation:

Step1: Divide numerator and denominator by \(x\)

$$\begin{align*} \lim_{x ightarrow\infty}\frac{4x}{5x - 2}&=\lim_{x ightarrow\infty}\frac{\frac{4x}{x}}{\frac{5x}{x}-\frac{2}{x}}\\ &=\lim_{x ightarrow\infty}\frac{4}{5-\frac{2}{x}} \end{align*}$$

Step2: Use the limit property \(\lim_{x

ightarrow\infty}\frac{a}{x}=0\) (\(a\) is a constant)
Since \(\lim_{x
ightarrow\infty}\frac{2}{x} = 0\), then \(\lim_{x
ightarrow\infty}\frac{4}{5-\frac{2}{x}}=\frac{4}{5 - 0}\)

Answer:

A. \(\lim_{x
ightarrow\infty}\frac{4x}{5x - 2}=\frac{4}{5}\)