Question

evaluate the limit.
lim (4x - 8)
simplify any fractions in your answer.

Explanation:

Step1: Analyze the behavior of \(4x\) as \(x\to-\infty\)

As \(x\) approaches \(-\infty\), the term \(4x\) also approaches \(-\infty\) (since \(4>0\) and \(x\to-\infty\)). Mathematically, \(\lim_{x\to-\infty}4x = -\infty\).

Step2: Analyze the constant term

The limit of a constant \(c\) as \(x\) approaches any value (including \(-\infty\)) is just the constant itself. So, \(\lim_{x\to-\infty}(- 8)=-8\).

Step3: Use the sum - rule for limits

The sum - rule for limits states that \(\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)\) (here \(a = -\infty\), \(f(x)=4x\), and \(g(x)=-8\)).
\(\lim_{x\to-\infty}(4x - 8)=\lim_{x\to-\infty}4x+\lim_{x\to-\infty}(-8)\)
Since \(\lim_{x\to-\infty}4x = -\infty\) and \(\lim_{x\to-\infty}(-8)=-8\), and \(-\infty-8=-\infty\)

Answer:

\(-\infty\)