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
sub in a number like 2.999
f(2.999) = |2.999-3|/(2.999-3) = .007/-.007 = -1
To find the left-hand limit of f(x) = |x-3|/(x-3) as x approaches 3 from the left, we need to evaluate the function as x gets closer and closer to 3 from values less than 3.
Let's start by substituting x = 3 - ε into the function, where ε is a very small positive number.
f(3 - ε) = |(3 - ε) - 3|/(3 - ε - 3)
Simplifying this, we get:
f(3 - ε) = | -ε | / (-ε)
As x approaches 3 from the left, ε approaches 0 from the positive side. So, we have:
f(3 - ε) = | -ε | / (-ε) = -ε / -ε = 1
Therefore, the left-hand limit of f(x) as x approaches 3 is 1.
Answer: A. 1
To find the left-hand limit of a function as x approaches a certain value, we need to evaluate the function for x values that are slightly less than the given value. In this case, we want to find the left-hand limit of f(x) = |x-3| / (x-3) as x approaches 3.
To start, let's consider x values approaching 3 from the left side, which means x values slightly less than 3.
Since the denominator (x-3) approaches 0 as x approaches 3, we need to determine the value of the numerator (|x-3|) as x approaches 3 from the left-hand side.
For x values less than 3 (x < 3), the expression |x-3| evaluates to -(x-3) as the negative sign of (x-3) will make it positive. So, |x-3| = -(x-3).
Now, we can substitute this value into the function and evaluate it:
f(x) = |x-3| / (x-3)
= -(x-3) / (x-3)
= -1.
Therefore, as x approaches 3 from the left-hand side, the function f(x) approaches -1.
Thus, the answer is D. -1.