To find the limit [tex]\(\lim_{x \to 0^-} f(x)\)[/tex], we need to evaluate the function [tex]\(f(x)\)[/tex] as [tex]\(x\)[/tex] approaches 0 from the left side (i.e., from negative values of [tex]\(x\)[/tex]).
The given piecewise function is:
[tex]\[ f(x) = \begin{cases}
\frac{x^2 - 16}{x + 4} & \text{if } x > 0, \\
\frac{x^2 - 16}{x - 4} & \text{if } x < 0.
\end{cases} \][/tex]
Since we are interested in the limit as [tex]\(x\)[/tex] approaches 0 from the negative side, we use the part of the function that applies to [tex]\(x < 0\)[/tex]:
[tex]\[ f(x) = \frac{x^2 - 16}{x - 4}. \][/tex]
Let's first factorize the numerator [tex]\(x^2 - 16\)[/tex]:
[tex]\[ x^2 - 16 = (x - 4)(x + 4). \][/tex]
So the expression becomes:
[tex]\[ \frac{(x - 4)(x + 4)}{x - 4}. \][/tex]
Now, as long as [tex]\(x \neq 4\)[/tex], the [tex]\((x - 4)\)[/tex] terms in the numerator and the denominator can be canceled out. Thus, the simplified function is:
[tex]\[ f(x) = x + 4. \][/tex]
Now, we need to find the limit of this simplified function as [tex]\(x\)[/tex] approaches 0 from the left:
[tex]\[ \lim_{x \to 0^-} (x + 4). \][/tex]
Substitute [tex]\(x = 0\)[/tex]:
[tex]\[ \lim_{x \to 0^-} (x + 4) = 0 + 4 = 4. \][/tex]
Therefore, the limit is:
[tex]\[ \lim_{x \to 0^-} f(x) = 4. \][/tex]
Thus, the answer is [tex]\(4\)[/tex].
