Q. 25 - Calculus

Question

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.  

$f\left(x\right)=\frac{\left(x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}$

Step-by-Step Solution

Answer

The root of $f\left(x\right)$ is $-1.$ $f\left(x\right)$ is discontinuous at $x=\pm 2.$ The horizontal asymptotes are at $y=1.$ The vertical asymptotes are at $x=-2.$

Step 1. Given information

The given function is $f\left(x\right)=\frac{\left(x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}.$ We need to determine the roots, discontinuous, horizontal, and vertical asymptotes of the function.

Step 2. Solution

$f\left(x\right)=\frac{\left(x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}.$

The root of the function is $x+1=0.$ 

                                                   $x=-1.$

The function is discontinuous at $x=\pm 2.$ By comparing the degree of the numerator and denominator horizontal asymptotes $y=1.$ The value of $x$ that cause the denominator of $f\left(x\right)$ to zero are $2.$ So the vertical asymptotes are $x=-2.$