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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 1

Lesson 6: Determining limits using algebraic properties of limits: direct substitution

# Limits of piecewise functions: absolute value

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.D (LO)
,
LIM‑1.D.1 (EK)
Analyzing the limit of |x-3|/(x-3) at x=3. When we have an absolute value, it's useful to treat the function as a piecewise function. Created by Sal Khan.

## Want to join the conversation?

• At , why is it that limits only exist when both sides approach the same value?
• For a good intuitive sense on why this is the case, use your favorite graphing utility and graph the function f(x) = 1 / x. The shape of this particular graph is called a hyperbola.

You might notice that the graph drops rapidly upon reaching as y value of 0 from the negative (or left hand side). You'll also notice that the graph rises rapidly as the y value approaches 0 from the left had side.

Since the graph contains a discontinuity (and a pretty major one at that), the limit of the function as x approaches 0 does not exist, because the 0+ and 0- limits are not equal.
• I don't understand why the x-3 in the numerator becomes negative when x<3. Anyone care to explain?
• You know absolute value of a real number x is non-negative value of x. x can be either negative, zero or positive. So when we take absolute value of x ,(i.e. |x|)
1. when x is zero (x=0), then |x| = |0| = 0
2. when x is positive (x>0), then |x| = positive value
3. when x is negative(x<0), then |x| = -1*x = positive value , so you are getting absolute value of a negative number x and in order to get non-negative same magnitude of x, you multiply negative value of x with -1 and you get positive value.

In the example of this video, you are taking absolute value of x-3, so if
1. x=3, you are getting |3-3|=0
2. x > 3, (x-3) is positive, so |x-3| positive
3. x < 3, (x-3) is negative, so |x-3| = -1*(x-3), which is positive
• isn't there a way to solve these limits without graphs and manual substitution method?
• Absolutely, that's what Calculus is all about :D
• Are "does not exist" and "it's undefined" the same thing?
• They are not. A limit of a function does not exist means it's limits from left and right aren't congruent. An example would be the floor function [x]. When you approach the same number (any point) from the left, you get a value x for it, but when you approach it from the right you get x+1. When it's undefined the limit can be calculated when you approach it from one side, but not the other. 1/x is an undefined function.
• For those wondering why the negative absolute value computation gives us a negative number (as I did), remember that there's a denominator that becomes negative! We're so focused on the numerator being an absolute value, we forget to look at the denominator. The numerator is indeed positive, but not the denominator.

In other words :
x>3 --> if x=4, f(4) = 1/1 = 1 (first case)
x<3 --> if x=2, F(2) = 1/-1 = -1 (second case)
• I know you can take limit from both positive or negative direction of real numbers but is it possible to approach a limit from the complex numbers plane ?
• For a real function, no, because the function is undefined in the imaginary domain. However, there are such things as complex limits for complex functions.
• why cant the limit be |1|?
• how about the absolute limit of f(x) as x approaches 3? that would be 1, no? (don't actually know if this'd be useful anywhere, but still)
(1 vote)
• Why do we say that the limit doesn't exist when the limits from both sides give us different answers? why can't there be two limits or just one limit?
• The limit of a function gives the value of the function as it gets infinitely closer to an x value. If the function approaches 4 from the left side of, say, x=-1, and 9 from the right side, the function doesn't approach any one number. The limit from the left and right exist, but the limit of a function can't be 2 y values.
• how do we define a 'limit' .
• The y value a function approaches when x approaches a certain value. If the y has different values as the x value is approached, then the limit doesn't exist. For example, in the video, as you looked at the graph from the left, you get a value of -1, and from the right, you get 1. So, as -1 is not equal to 1, there is no limit. If they approach the same y value from both sides, the limit does exist, and is is equivalent to the y value approached.