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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.

Find limit as x approaches 2 from the left of the quotient of the absolute value of-example-1
User KManish
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6 votes

Answer:


\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.

User Mayank Kataria
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