To evaluate the limit
[tex]\[
\lim _{x \rightarrow-6} \left(\frac{1 - 8x}{1 + 8x}\right)^3,
\][/tex]
we can break it down into simpler steps.
1. Substitute [tex]\( x = -6 \)[/tex] into the function:
[tex]\[
\left(\frac{1 - 8(-6)}{1 + 8(-6)}\right)^3.
\][/tex]
2. Simplify the numerator and denominator separately:
- For the numerator:
[tex]\[
1 - 8(-6) = 1 + 48 = 49.
\][/tex]
- For the denominator:
[tex]\[
1 + 8(-6) = 1 - 48 = -47.
\][/tex]
3. Combine the numerator and denominator:
[tex]\[
\left( \frac{49}{-47} \right)^3 = \left( -\frac{49}{47} \right)^3.
\][/tex]
4. Simplify the expression:
[tex]\[
\left( -\frac{49}{47} \right)^3 = -\left( \frac{49}{47} \right)^3 = -\frac{49^3}{47^3}.
\][/tex]
Now we evaluate the powers in the fraction:
[tex]\[
49^3 = 49 \times 49 \times 49 = 117649,
\][/tex]
and
[tex]\[
47^3 = 47 \times 47 \times 47 = 103823.
\][/tex]
5. Combine the results:
[tex]\[
- \frac{117649}{103823}.
\][/tex]
Therefore, the limit is
[tex]\[
\boxed{-\frac{117649}{103823}}.
\][/tex]
