Q. 20 - Calculus

Question

Evaluate the limit of the expression

$\underset{x\to \infty }{\lim }\frac{(x-1)(3x+1{)}^3}{(x-2{)}^4}$

Step-by-Step Solution

Answer

The limit  of the expression is 0.

Step 1. Given

$\underset{x\to \infty }{\lim }\frac{(x-1)(3x+1{)}^3}{(x-2{)}^4}$The given expression is 

Step 2. Calculation

The evaluated expression is 

$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{(x-1)(3x+1{)}^3}{(x-2{)}^4} \\ =\underset{x\to \infty }{\lim }\frac{x\left(1-\frac{1}{x}\right)x^3\left({\displaystyle \frac{3}{x^2}}+\frac{1}{x^3}\right)}{x^4\left(1-\frac{2}{x^4}\right)} \\ =\underset{x\to \infty }{\lim }\frac{\left(1-\frac{1}{x}\right)\left(\frac{3}{x^2}+\frac{1}{x^3}\right)}{\left(1-\frac{2}{x^4}\right)} \\ =\frac{\left(1-\frac{1}{\infty }\right)\left({\displaystyle \frac{3}{\infty }}+\frac{1}{\infty }\right)}{\left(1-\frac{2}{{\infty }^4}\right)} \\ =\frac{(1-0)(0+0)}{(1-0)} \\ =0 \end{gathered}$$