What is the left-hand limit of f(x)=|x−3|/x−3 as x approaches 3?
A) 1
B) 2
C) 0
D) -1
E) -2
as x->3 from the left, (x-3) < 0, so |x-3| = -(x-3).
-(x-3)/(x-3) = -1
To find the left-hand limit of a function as x approaches a certain value, we evaluate the function from the left side of that value. In this case, we want to find the left-hand limit of f(x)=|x−3|/(x−3) as x approaches 3.
Let's first consider the function f(x)=|x−3|/(x−3) and see how it behaves for x values slightly less than 3. We can plug in some values to observe the behavior.
For x=2.9:
f(2.9) = |2.9−3|/(2.9−3) = 0.1/-0.1 = -1
For x=2.99:
f(2.99) = |2.99−3|/(2.99−3) = 0.01/-0.01 = -1
As x approaches 3 from the left side, the function seems to approach -1.
Therefore, the left-hand limit of f(x)=|x−3|/(x−3) as x approaches 3 is -1. So, the correct answer is (D) -1.