find the limit. write ∞ or - ∞ where appropriate.\nlim_{x→1} \\frac{-4}{(x - 1)^2}\nlim_{x→1} \\frac{-4}{(x…

find the limit. write ∞ or - ∞ where appropriate.\nlim_{x→1} \\frac{-4}{(x - 1)^2}\nlim_{x→1} \\frac{-4}{(x - 1)^2}=\\square (simplify your answer.)

find the limit. write ∞ or - ∞ where appropriate.\nlim_{x→1} \\frac{-4}{(x - 1)^2}\nlim_{x→1} \\frac{-4}{(x - 1)^2}=\\square (simplify your answer.)

Answer

Explanation:

Step1: Analyze the denominator as x approaches 1

Let (t=(x - 1)). As (x\rightarrow1), then (t\rightarrow0). The limit becomes (\lim_{t\rightarrow0}\frac{- 4}{t^{2}}).

Step2: Consider the sign and value of the denominator

When (t\rightarrow0), (t^{2}>0) for (t\neq0). And we have a constant - 4 in the numerator.

Step3: Determine the limit value

Since the numerator is negative (-4) and the denominator (t^{2}) approaches 0 from the positive - side as (t\rightarrow0), the value of the fraction (\frac{-4}{t^{2}}) approaches (-\infty).

Answer:

(-\infty)