Question

the graph of the function $c(x) = \frac{-3x^2 - 18x + 21}{x^2 + 8x + 7}$ is shown below. write a limit to represent the funct behavior as $x$ gets infinitely close to $-7$ from either side.
$x = -7.1$ $y = -3.9836$
$x = -7$ corresponds to the hole at $(-7, -4)$
$y$-values get closer and closer to $-4$
for a hole:
$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = \lim_{x \to a} f(x) = l$
from either side
$\lim_{x \to -7} (c(x)) = -4$

Explanation:

Step1: Identify the limit context

The problem is about the limit of the function \( c(x) = \frac{-3x^2 - 18x + 21}{x^2 + 8x + 7} \) as \( x \) approaches -7 from either side. We know from the given information that \( x = -7 \) corresponds to a hole at \( (-7, -4) \), and for a hole, the limit from the left, right, and overall exist and are equal.

Step2: Determine the limit value

The y - values get closer and closer to -4 as \( x \) approaches -7 from either side. Using the definition for the limit at a hole (where the left - hand limit, right - hand limit, and the overall limit are equal), we can write the limit of \( c(x) \) as \( x \) approaches -7.

The limit notation for the function \( c(x) \) as \( x \) approaches -7 (from either side) is \( \lim_{x
ightarrow - 7}c(x) \), and since the y - values approach -4, this limit is equal to -4.

Answer:

\(\lim_{x
ightarrow - 7}c(x)=-4\)