Q. 6 - Calculus

Question

Calculating limits: Find each limit by hand.

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

Step-by-Step Solution

Answer

The answer of the limit is 0.

Step 1. Given Information.

Given: $\underset{x\to -\infty }{\lim }\frac{x^3+2x+1}{1-x^4}$.

Step 2. Calculating the limit using direct substitution.

$$\begin{gathered} Solving by hand, \\ \underset{x\to -\infty }{\lim }\frac{x^3+2x+1}{1-x^4} \\ Dividing both numerator and denominator by x^4. \\ =\underset{x\to -\infty }{\lim }\frac{{\displaystyle \frac{1}{x}}+{\displaystyle \frac{2}{x^3}}+{\displaystyle \frac{1}{x^4}}}{{\displaystyle \frac{1}{x^4}}-1} \\ =\frac{{\displaystyle \frac{1}{\infty }}+{\displaystyle \frac{2}{\infty }}+{\displaystyle \frac{1}{\infty }}}{{\displaystyle \frac{1}{\infty }}-1} = \frac{0+0+0}{0-1}=0. \end{gathered}$$