find the limit. write ∞ or - ∞ where appropriate.\nlim(x→4) 1/(x - 4)^2\nlim(x→4) 1/(x - 4)^2 = (simplify…

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

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

Answer

Explanation:

Step1: Analyze the denominator as x approaches 4

Let (t=x - 4). As (x\rightarrow4), then (t\rightarrow0). The limit becomes (\lim_{t\rightarrow0}\frac{1}{t^{2}}).

Step2: Evaluate the limit

When (t) approaches 0 from either the left - hand side ((t\rightarrow0^{-})) or the right - hand side ((t\rightarrow0^{+})), (t^{2}) is a positive quantity approaching 0. So, (\frac{1}{t^{2}}) approaches positive infinity.

Answer:

(\infty)