Question

find the limit. write ∞ or - ∞ where appropriate.
lim(x→4) 1/(x - 4)^2
lim(x→4) 1/(x - 4)^2 = (simplify your answer.)

Explanation:

Step1: Analyze the denominator as x approaches 4

Let \(t=x - 4\). As \(x
ightarrow4\), then \(t
ightarrow0\). The limit becomes \(\lim_{t
ightarrow0}\frac{1}{t^{2}}\).

Step2: Evaluate the limit

When \(t\) approaches 0 from either the left - hand side (\(t
ightarrow0^{-}\)) or the right - hand side (\(t
ightarrow0^{+}\)), \(t^{2}\) is a positive quantity approaching 0. So, \(\frac{1}{t^{2}}\) approaches positive infinity.

Answer:

\(\infty\)