Question

find the limits in a) through c) below for the function (f(x)=\frac{x^{2}-11x + 10}{x + 10}). use (-infty) and (infty) when appropriate.
a) (lim_{x
ightarrow - 10^{-}}) (simplify your answer.)
a. the limit does not exist and is neither (-infty) nor (infty).
b) select the correct choice below and fill in any answer boxes in your choice.
a. (lim_{x
ightarrow - 10^{+}}f(x)=infty) (simplify your answer.)
b. the limit does not exist and is neither (-infty) nor (infty).
c) select the correct choice below and fill in any answer boxes in your choice.
a. (lim_{x
ightarrow - 10}f(x)=infty) (simplify your answer.)
b. the limit does not exist and is neither (-infty) nor (infty).

Explanation:

Step1: Factor the numerator

First, factor the numerator $x^{2}-11x + 10$. We have $x^{2}-11x + 10=(x - 10)(x - 1)$ using the formula for factoring quadratic $ax^{2}+bx + c=a(x - x_1)(x - x_2)$ where $x_1,x_2$ are the roots of $ax^{2}+bx + c = 0$. Here, for $x^{2}-11x + 10=0$, $x=\frac{11\pm\sqrt{121 - 40}}{2}=\frac{11\pm9}{2}$, so the roots are $x = 10$ and $x = 1$. So $f(x)=\frac{(x - 10)(x - 1)}{x + 10}$.

Step2: Analyze the limit as $x\to - 10^{+}$

As $x\to - 10^{+}$, the numerator $(x - 10)(x - 1)\to(-10 - 10)(-10 - 1)=(-20)\times(-11)=220$ (a non - zero constant), and the denominator $x + 10\to0^{+}$ (a positive infinitesimal). Then $\lim_{x\to - 10^{+}}\frac{(x - 10)(x - 1)}{x + 10}=\infty$.

Step3: Analyze the limit as $x\to - 10^{-}$

As $x\to - 10^{-}$, the numerator $(x - 10)(x - 1)\to(-10 - 10)(-10 - 1)=220$ (a non - zero constant), and the denominator $x + 10\to0^{-}$ (a negative infinitesimal). Then $\lim_{x\to - 10^{-}}\frac{(x - 10)(x - 1)}{x + 10}=-\infty$. Since the left - hand limit and the right - hand limit are not equal, the two - sided limit $\lim_{x\to - 10}f(x)$ does not exist.

Answer:

a) The question seems incomplete as the limit value for part a) is not clearly stated.
b) A. $\lim_{x\to - 10^{+}}f(x)=\infty$
c) The limit does not exist and is neither $-\infty$ nor $\infty$ since $\lim_{x\to - 10^{-}}f(x)=-\infty$ and $\lim_{x\to - 10^{+}}f(x)=\infty$.