Questions: Find the limits in a) through c) below for the function f(x) = (x^2 - 11x + 10) / (x + 10). Use -∞ and ∞ when appropriate. A. lim f(x) = -∞ x -> -10^- (Simplify your answer.) B. The limit does not exist and is neither -∞ nor ∞. b) Select the correct choice below and fill in any answer boxes in your choice. A. lim x -> -10^+ f(x) = ∞ x -> -10^+ (Simplify your answer.) B. The limit does not exist and is neither -∞ nor ∞. c) Select the correct choice below and fill in any answer boxes in your choice. A. lim x -> -10 f(x) = (Simplify your answer.)

Find the limits in a) through c) below for the function f(x) = (x^2 - 11x + 10) / (x + 10). Use -∞ and ∞ when appropriate.
A. lim f(x) = -∞
x -> -10^-
(Simplify your answer.)
B. The limit does not exist and is neither -∞ nor ∞.
b) Select the correct choice below and fill in any answer boxes in your choice.
A. lim x -> -10^+ f(x) = ∞
x -> -10^+
(Simplify your answer.)
B. The limit does not exist and is neither -∞ nor ∞.
c) Select the correct choice below and fill in any answer boxes in your choice.
A. lim x -> -10 f(x) =
(Simplify your answer.)

Transcript text: Find the limits in a ) through c ) below for the function $f(x)=\frac{x^{2}-11 x+10}{x+10}$. Use $-\infty$ and $\infty$ when appropriate. A. $\quad \lim f(x)=-\infty$ $\square$ \[ x \rightarrow-10^{-} \] (Simplify your answer.) B. 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. $\lim _{x \rightarrow-10^{+}} f(x)=\infty$ \[ x \rightarrow-10^{+} \] $\square$ (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. $\lim _{x \rightarrow-10} f(x)=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

To find the limits of the function $ f(x) = \frac{x^2 - 11x + 10}{x + 10} $ as $ x $ approaches specific values, we need to analyze the behavior of the function around these points.

A. As $ x \to -10^- $, we approach -10 from the left. We need to check the sign of the numerator and denominator to determine if the function approaches $-\infty$ or $\infty$.

B. As $ x \to -10^+ $, we approach -10 from the right. Again, we check the sign of the numerator and denominator to determine the behavior of the function.

C. As $ x \to -10 $, we need to determine if the limit exists by checking if the left-hand limit and right-hand limit are equal.

Step 1: Analyze the Function

The function given is $ f(x) = \frac{x^2 - 11x + 10}{x + 10} $. We need to find the limits as $ x $ approaches $-10$ from different directions.

Step 2: Limit as $ x \to -10^- $

As $ x $ approaches $-10$ from the left ($ x \to -10^- $), the denominator $ x + 10 $ approaches zero from the negative side. The numerator $ x^2 - 11x + 10 $ evaluates to a positive number near $ x = -10 $. Therefore, the function approaches $-\infty$.

Step 3: Limit as $ x \to -10^+ $

As $ x $ approaches $-10$ from the right ($ x \to -10^+ $), the denominator $ x + 10 $ approaches zero from the positive side. The numerator remains positive near $ x = -10 $. Therefore, the function approaches $\infty$.

Step 4: Limit as $ x \to -10 $

The limit as $ x \to -10 $ does not exist because the left-hand limit and the right-hand limit are not equal. Specifically, the left-hand limit is $-\infty$ and the right-hand limit is $\infty$.