QuestionFind $\lim _{x \rightarrow 2} f(x)$ for the function $f(x)=\frac{x^{2}-4}{x^{2}+x-6}$. Choices: (A) 0, (B) $\frac{2}{3}$, (C) $\frac{4}{5}$, (D) nonexistent.

Studdy Solution
Assumptions1. The function is defined as $f(x)=\frac{x^{}-4}{x^{}+x-6}$ . We need to find the limit of the function as $x$ approachesFirst, we will try to substitute $x=$ directly into the function to see if it's defined at this point.
$$f()=\frac{^{}-4}{^{}+-6}$$3Calculate the value of the function at $x=$.
$$f()=\frac{4-4}{4+-6}$$4implify the expression.
$$f()=\frac{0}{0}$$5As we can see, the function is undefined at $x=$, which is an indeterminate form. So, we need to simplify the function to find the limit as $x$ approaches.
6The function can be simplified by factoring the numerator and the denominator.
$$f(x)=\frac{(x-)(x+)}{(x-)(x+3)}$$7After factoring, we can cancel out the common factor $(x-)$ from the numerator and the denominator.
$$f(x)=\frac{x+}{x+3}$$8Now, we can substitute $x=$ into the simplified function.
$$f()=\frac{+}{+3}$$9Calculate the value of the function at $x=$.
$$f()=\frac{4}{5}$$Therefore, $\lim{x \rightarrow} f(x) = \frac{4}{5}$.