Chapter 13.1, Problem 21E

To determine

To Get: The limit value of given expression at x tends to $1$

Answer to Problem 21E

The limit value of given expression at x tends to $1$ is $-8$

Explanation of Solution

Given information:

Function expression $\frac{f\left(x\right)}{g\left(x\right)}$ is given at $x\to 1$

Concept used:

Use the Hospital’s Rule to evaluate the in determinate limits

  $\underset{x\to d}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to d}{\lim }\frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}$

By observing $f\left(x\right)=x^3+x^2+3x-5$ and $g\left(x\right)=2x^2-5x+3$

First substitute $x=1$ in the denominator if denominator value is $0$ then use the L Hospital’s Rule to evaluate.

  $\begin{array}{l}\Rightarrow \underset{x\to 1}{\lim }\frac{x^3+x^2+3x-5}{2x^2-5x+3}\\ \Rightarrow \underset{x\to 1}{\lim }\frac{3x^2+2x+3}{4x-5}\left(\text{by using L'Hospital Rule}\right)\\ \Rightarrow \frac{3+2+3}{-1}\\ \Rightarrow -8\end{array}$

Hence, The limit value of given expression at x tends to $1$ is $-8$