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

`\lim_(x \to 2^-) (|x-2|)/(x-2)=-1`.

Explanation:

We want to find
`\lim_(x \to 2^-) (|x-2|)/(x-2)`.

By definition:

`|x-2|=\left \{ {{x-2,\:if\:x\:>\:2} \atop {-(x-2),\:if\:x\:<\:2}} \right.`

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

`\implies \lim_(x \to 2^-) (|x-2|)/(x-2)=\lim_(x \to 2) (-(x-2))/(x-2)`.

`\implies \lim_(x \to 2^-) (|x-2|)/(x-2)=\lim_(x \to 2) (-1)`.

The limit of a constant is the constant.

`\implies \lim_(x \to 2^-) (|x-2|)/(x-2)=-1`.