To find the limit of the function [tex]\(\lim_{x \rightarrow 6} \frac{x^2-36}{x-6}\)[/tex], follow these steps:
1. Identify the Indeterminate Form:
First, notice that direct substitution of [tex]\( x = 6 \)[/tex] into the function [tex]\(\frac{x^2-36}{x-6}\)[/tex]:
[tex]\[
\frac{6^2 - 36}{6 - 6} = \frac{36 - 36}{0} = \frac{0}{0}
\][/tex]
This is an indeterminate form, so we need to simplify the expression.
2. Simplify the Expression:
Factor the numerator [tex]\(x^2 - 36 \)[/tex]. Recognize that [tex]\( x^2 - 36 \)[/tex] is a difference of squares:
[tex]\[
x^2 - 36 = (x - 6)(x + 6)
\][/tex]
Substituting this back into the limit expression, we get:
[tex]\[
\lim_{x \rightarrow 6} \frac{(x - 6)(x + 6)}{x - 6}
\][/tex]
We can cancel out the [tex]\(x - 6\)[/tex] in the numerator and denominator:
[tex]\[
\lim_{x \rightarrow 6} (x + 6)
\][/tex]
3. Evaluate the Simplified Expression:
Now, after canceling, the limit simplifies to:
[tex]\[
\lim_{x \rightarrow 6} (x + 6)
\][/tex]
Substitute [tex]\(x = 6\)[/tex] into the simplified expression:
[tex]\[
6 + 6 = 12
\][/tex]
Thus, the limit is:
[tex]\[\boxed{12}\][/tex]
