Question

find the limit. write ∞ or - ∞ where appropriate.
lim(x→4⁻) 1/(x - 4)
lim(x→4⁻) 1/(x - 4)=□ (simplify your answer.)

Explanation:

Step1: Analyze the left - hand limit

As \(x\to4^{-}\), it means \(x\) approaches \(4\) from the left - hand side. So \(x<4\), and then \(x - 4<0\). Let \(t=x - 4\), as \(x\to4^{-}\), \(t\to0^{-}\).

Step2: Evaluate the limit

We have \(\lim_{x\to4^{-}}\frac{1}{x - 4}=\lim_{t\to0^{-}}\frac{1}{t}\). When \(t\) approaches \(0\) from the left - hand side, \(\frac{1}{t}\) approaches \(-\infty\).

Answer:

\(-\infty\)