Question

Find limit as x approaches 2 from the left of the quotient of the absolute value of the quantity x minus 2, and the quantity x minus 2. You must show your work or explain your work in words.

Answer 1

Answer:

$\lim_{x \to 2^-} \frac{|x-2|}{x-2}=-1$.

Step-by-step explanation:

We want to find $\lim_{x \to 2^-} \frac{|x-2|}{x-2}$.

By definition:

$|x-2|=\left \{ {{x-2,\:if\:x\:>\:2} \atop {-(x-2),\:if\:x\:$

Since we want to find the Left Hand Limit, we use f(x)=-(x-2)

$\implies \lim_{x \to 2^-} \frac{|x-2|}{x-2}=\lim_{x \to 2} \frac{-(x-2)}{x-2}$.

$\implies \lim_{x \to 2^-} \frac{|x-2|}{x-2}=\lim_{x \to 2} (-1)$.

The limit of a constant is the constant.

$\implies \lim_{x \to 2^-} \frac{|x-2|}{x-2}=-1$.