select the correct answer.\nwhat is the difference between the right - hand limit and the left - hand limit…

select the correct answer.\nwhat is the difference between the right - hand limit and the left - hand limit of $f(x)=\frac{|x - 3|}{x - 3}$ as x approaches 3?\na. no difference\nb. 1 unit\nc. 2 units\nd. 3 units

select the correct answer.\nwhat is the difference between the right - hand limit and the left - hand limit of $f(x)=\frac{|x - 3|}{x - 3}$ as x approaches 3?\na. no difference\nb. 1 unit\nc. 2 units\nd. 3 units

Answer

Explanation:

Step1: Define the absolute - value function

For (x\geq3), (|x - 3|=x - 3); for (x<3), (|x - 3|=-(x - 3)).

Step2: Calculate the right - hand limit ((x\to3^{+}))

When (x\to3^{+}), (x>3), so (f(x)=\frac{|x - 3|}{x - 3}=\frac{x - 3}{x - 3}=1). Then (\lim_{x\to3^{+}}f(x)=1).

Step3: Calculate the left - hand limit ((x\to3^{-}))

When (x\to3^{-}), (x<3), so (f(x)=\frac{|x - 3|}{x - 3}=\frac{-(x - 3)}{x - 3}=-1). Then (\lim_{x\to3^{-}}f(x)=-1).

Step4: Find the difference

The difference between the right - hand limit and the left - hand limit is (1-(-1)=2).

Answer:

C. 2 units