Question

question 13 evaluate the limit: lim(x→2⁻) (x - 5)/(x - 2)². ∞ -5/3 no correct answer choice is given. -∞ 5/3

Explanation:

Step1: Analyze the denominator

As $x\to2^{-}$, let $t=x - 2$, then $t\to0^{-}$. The denominator $(x - 2)^2=t^2$. And $t^2>0$ when $t
eq0$. As $t\to0^{-}$, $t^2\to0^{+}$.

Step2: Analyze the numerator

When $x\to2^{-}$, the numerator $x - 5\to2-5=-3$.

Step3: Evaluate the limit

We have $\lim_{x\to2^{-}}\frac{x - 5}{(x - 2)^2}=\lim_{t\to0^{-}}\frac{t - 3}{t^2}$. Since the numerator approaches - 3 and the denominator approaches $0^{+}$, the value of the fraction is $-\infty$.

Answer:

D. $-\infty$