Let's evaluate the limit
[tex]\[
\lim _{x \rightarrow 8} \frac{3 x^2 - 9 x - 162}{x^2 + 11 x + 30}.
\][/tex]
First, we need to examine the function given in the limit:
[tex]\[
f(x) = \frac{3 x^2 - 9 x - 162}{x^2 + 11 x + 30}.
\][/tex]
Evaluate the numerator and denominator at [tex]\( x = 8 \)[/tex]:
[tex]\[
\text{Numerator} = 3(8)^2 - 9(8) - 162 = 192 - 72 - 162 = -42,
\][/tex]
[tex]\[
\text{Denominator} = (8)^2 + 11(8) + 30 = 64 + 88 + 30 = 182.
\][/tex]
Thus, the function at [tex]\( x = 8 \)[/tex] is:
[tex]\[
f(8) = \frac{-42}{182}.
\][/tex]
Next, we shall simplify this fraction:
[tex]\[
\frac{-42}{182} = -\frac{42}{182}.
\][/tex]
To find the simplest form, we need to determine the greatest common divisor (GCD) of 42 and 182. The GCD of 42 and 182 is 14, so we have:
[tex]\[
- \frac{42 \div 14}{182 \div 14} = -\frac{3}{13}.
\][/tex]
Thus,
[tex]\[
f(8) = -\frac{3}{13}.
\][/tex]
Finally, let's round this result to two decimal places. Converting [tex]\(-\frac{3}{13}\)[/tex] to its decimal form:
[tex]\[
- \frac{3}{13} \approx -0.230769230769\ldots \approx -0.23.
\][/tex]
Therefore, the limit is:
[tex]\[
\lim _{x \rightarrow 8} \frac{3 x^2 - 9 x - 162}{x^2 + 11 x + 30} = -\frac{3}{13},
\][/tex]
which rounds to two decimal places as:
[tex]\[
-0.23.
\][/tex]
So, the final answer is:
[tex]\[
-\frac{3}{13}, -0.23.
\][/tex]
