To evaluate the limit
[tex]\[
\lim _{x \rightarrow 2}\left(\frac{1-2 x}{1+6 x}\right)^4,
\][/tex]
we need to carefully analyze the behavior of the expression as [tex]\( x \)[/tex] approaches 2.
### Step-by-Step Solution:
1. Identify the function inside the limit:
[tex]\[
f(x) = \left( \frac{1 - 2x}{1 + 6x} \right)^4.
\][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[
f(2) = \left( \frac{1 - 2 \cdot 2}{1 + 6 \cdot 2} \right)^4.
\][/tex]
3. Simplify the numerator and the denominator separately:
[tex]\[
\text{Numerator: } 1 - 2 \cdot 2 = 1 - 4 = -3.
\][/tex]
[tex]\[
\text{Denominator: } 1 + 6 \cdot 2 = 1 + 12 = 13.
\][/tex]
4. Substitute the simplified values back into the function:
[tex]\[
f(2) = \left( \frac{-3}{13} \right)^4.
\][/tex]
5. Simplify the fraction raised to the power of 4:
[tex]\[
\left( \frac{-3}{13} \right)^4 = \left( \frac{3}{13} \right)^4.
\][/tex]
Note that raising a negative number to an even power results in a positive number.
6. Calculate the power:
[tex]\[
\left( \frac{3}{13} \right)^4 = \frac{3^4}{13^4}.
\][/tex]
7. Compute the final values:
[tex]\[
3^4 = 81,
\][/tex]
[tex]\[
13^4 = 28561.
\][/tex]
Combining these results, we find that:
[tex]\[
\left( \frac{3}{13} \right)^4 = \frac{81}{28561}.
\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{x \rightarrow 2}\left(\frac{1-2 x}{1+6 x}\right)^4 = \frac{81}{28561}.
\][/tex]
