Q. 19 - Calculus

Question

Consider the limit expression

$\underset{x\to \infty }{\lim }\frac{(2x-1)\left(x^2+1\right)}{x^2-4}$

Calculate the limit.

Step-by-Step Solution

Answer

The limit  of the expression is $\infty$.

Step 1. Given

The given expression is 

$\underset{x\to \infty }{\lim }\frac{(2x-1)\left(x^2+1\right)}{x^2-4}$

Step 2. Calculation

The evaluated expression is 

$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{(2x-1)\left(x^2+1\right)}{x^2-4} \\ =\underset{x\to \infty }{\lim }\frac{x\left(2-\frac{1}{x}\right)x\left(x+\frac{1}{x}\right)}{x^2\left(1-\frac{4}{x^2}\right)} \\ =\underset{x\to \infty }{\lim }\frac{\left(2-\frac{1}{x}\right)\left(x+\frac{1}{x}\right)}{\left(1-\frac{4}{x^2}\right)} \\ =\frac{\left(2-\frac{1}{\infty }\right)\left(\infty +\frac{1}{\infty }\right)}{\left(1-\frac{4}{{\infty }^2}\right)} \\ =\frac{(2-0)(\infty +0)}{(1-0)} \\ =\frac{2\infty }{1} \\ =\infty \end{gathered}$$