use direct substitution to obtain an undefined expression. then, use the method of evaluating a limit of the…

use direct substitution to obtain an undefined expression. then, use the method of evaluating a limit of the form $k/0, k\neq0$ to simplify the function to help determine the limit. $lim_{x\rightarrow - 2^{-}}\frac{2x^{2}+9x - 5}{x^{2}+x - 2}$

use direct substitution to obtain an undefined expression. then, use the method of evaluating a limit of the form $k/0, k\neq0$ to simplify the function to help determine the limit. $lim_{x\rightarrow - 2^{-}}\frac{2x^{2}+9x - 5}{x^{2}+x - 2}$

Answer

Explanation:

Step1: Direct substitution

Substitute $x = - 2$ into $\frac{2x^{2}+9x - 5}{x^{2}+x - 2}$. $2(-2)^{2}+9(-2)-5=2\times4-18 - 5=8-18 - 5=-15$. $(-2)^{2}+(-2)-2=4-2 - 2=0$. So we get $\frac{-15}{0}$, an undefined expression.

Step2: Factor the numerator and denominator

Factor $2x^{2}+9x - 5=(2x - 1)(x + 5)$. Factor $x^{2}+x - 2=(x + 2)(x - 1)$. So the function becomes $\frac{(2x - 1)(x + 5)}{(x + 2)(x - 1)}$.

Step3: Analyze the limit

As $x\to - 2$, the limit $\lim_{x\to - 2}\frac{2x^{2}+9x - 5}{x^{2}+x - 2}=\lim_{x\to - 2}\frac{(2x - 1)(x + 5)}{(x + 2)(x - 1)}$. The numerator approaches $(2(-2)-1)(-2 + 5)=(-4 - 1)\times3=-15$. The denominator approaches $(-2 + 2)(-2 - 1)=0$. Since the numerator approaches a non - zero value $(-15)$ and the denominator approaches $0$, the limit $\lim_{x\to - 2}\frac{2x^{2}+9x - 5}{x^{2}+x - 2}$ does not exist.

Answer:

The limit does not exist.