Q. 5 - Calculus

Question

Calculating limits: Find each limit by hand.

$\underset{x\to 1}{\lim }\frac{2x^3-x^2-2x+1}{x^2-2x+1}$.

Step-by-Step Solution

Answer

The answer of the limit is $\infty$.

Step 1. Given Information.

Given: $\underset{x\to 1}{\lim }\frac{2x^3-x^2-2x+1}{x^2-2x+1}$.

Step 2. Calculating the limit using direct substitution.

$$\begin{gathered} Solving by hand, \\ \underset{x\to 1}{\lim }\frac{2x^3-x^2-2x+1}{x^2-2x+1} \\ = \underset{x\to 1}{\lim }\frac{(x-1)(x+1)(2x-1)}{{(x-1)}^2} \\ =\underset{x\to 1}{\lim }\frac{(x+1)(2x-1)}{(x-1)} = \frac{2\times 1}{0} = \infty . \end{gathered}$$