Question

evaluate the limit.

lim (-2x - 7)
x→∞

simplify any fractions in your answer.

Explanation:

Step1: Analyze the limit of \(-2x\) as \(x\to\infty\)

As \(x\to\infty\), for the function \(y = - 2x\), when \(x\) gets larger and larger, \(y=-2x\to-\infty\) (since the coefficient of \(x\) is negative). Mathematically, \(\lim_{x\to\infty}(-2x)=-\infty\).

Step2: Analyze the limit of the constant term

For a constant \(c\), \(\lim_{x\to\infty}c = c\). Here \(c=-7\), so \(\lim_{x\to\infty}(-7)=-7\).

Step3: Use the sum rule for limits

The sum rule for limits states that \(\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)\) (when both limits on the right - hand side exist in the extended real - number system).
For \(f(x)=-2x\) and \(g(x)=-7\), \(\lim_{x\to\infty}(-2x - 7)=\lim_{x\to\infty}(-2x)+\lim_{x\to\infty}(-7)\)
Since \(\lim_{x\to\infty}(-2x)=-\infty\) and \(\lim_{x\to\infty}(-7)=-7\), then \(\lim_{x\to\infty}(-2x - 7)=-\infty\)

Answer:

\(-\infty\)