3. (15 points) Find the value(s) of $a$ so that the limit is true.
\[
\lim _{x \rightarrow-\infty} \frac{(x-2)\left(3 x^{2}+5\right)-4 a x^{3}}{8 a x^{3}+5 x-1}=1
\]
Answer
Final Answer: The value of \(a\) that makes the limit true is \(\boxed{\frac{1}{4}}\).
Step 1 ：The limit is given as \(x\) approaches \(-\infty\). This means that the highest degree terms in the numerator and denominator will dominate the value of the limit. Therefore, we can simplify the limit by dividing both the numerator and the denominator by \(x^{3}\), the highest power of \(x\) in the denominator. This will give us a simpler limit to solve for \(a\).
Step 2 ：By simplifying the limit, we get \(-\frac{4a - 3}{8a}\). We want this limit to be equal to 1. Therefore, we can set up the equation \(-\frac{4a - 3}{8a} = 1\) and solve for \(a\).
Step 3 ：Solving the equation gives us the value of \(a\) as \(\frac{1}{4}\).
Step 4 ：Final Answer: The value of \(a\) that makes the limit true is \(\boxed{\frac{1}{4}}\).
