To find the values of the limits [tex]\(\lim_{x \to -6} f(x)\)[/tex], [tex]\(\lim_{x \to -2} f(x)\)[/tex], and [tex]\(\lim_{x \to 4} f(x)\)[/tex], we need to systematically evaluate each one as [tex]\(x\)[/tex] approaches the given points.
1. Evaluate [tex]\(\lim_{x \to -6} f(x)\)[/tex]:
- As [tex]\(x\)[/tex] approaches [tex]\(-6\)[/tex], we need to determine the behavior of [tex]\(f(x)\)[/tex] near this point.
- The limit essentially captures the value that [tex]\(f(x)\)[/tex] converges to as [tex]\(x\)[/tex] gets arbitrarily close to [tex]\(-6\)[/tex].
Therefore,
[tex]\[
\lim_{x \to -6} f(x) = f(-6)
\][/tex]
2. Evaluate [tex]\(\lim_{x \to -2} f(x)\)[/tex]:
- Similarly, as [tex]\(x\)[/tex] approaches [tex]\(-2\)[/tex], we must examine [tex]\(f(x)\)[/tex] in the vicinity of [tex]\(-2\)[/tex].
- We are interested in the value that [tex]\(f(x)\)[/tex] reaches as [tex]\(x\)[/tex] gets very close to [tex]\(-2\)[/tex].
Hence,
[tex]\[
\lim_{x \to -2} f(x) = f(-2)
\][/tex]
3. Evaluate [tex]\(\lim_{x \to 4} f(x)\)[/tex]:
- Lastly, as [tex]\(x\)[/tex] approaches [tex]\(4\)[/tex], we analyze [tex]\(f(x)\)[/tex] around this point.
- This limit represents the value [tex]\(f(x)\)[/tex] approaches as [tex]\(x\)[/tex] gets very close to [tex]\(4\)[/tex].
Thus,
[tex]\[
\lim_{x \to 4} f(x) = f(4)
\][/tex]
Summarizing our findings,
- [tex]\(\lim_{x \to -6} f(x)\)[/tex] is given by [tex]\(f(-6)\)[/tex]
- [tex]\(\lim_{x \to -2} f(x)\)[/tex] is given by [tex]\(f(-2)\)[/tex]
- [tex]\(\lim_{x \to 4} f(x)\)[/tex] is given by [tex]\(f(4)\)[/tex]
Thus, the evaluated limits are:
[tex]\[
\lim_{x \to -6} f(x) = f(-6), \quad \lim_{x \to -2} f(x) = f(-2), \quad \lim_{x \to 4} f(x) = f(4)
\][/tex]
