Q: What is limx→2 (x-2)/|x-2|
A: This question is not too bad if you know what the function (x-2)/|x-2| looks like graphically. But, let’s say you don’t.
We are going to “talk our way” through this problem to help solidify the concept of a limit.
If you plug in 2 to the function, you are finding the value of the function when x=2. This is important, and related, though it is not the limit. This is even sometimes a skill used to help us find the limit, but it is still not the limit. Sometimes the function value is equal to the function limit, which can also be confusing, but not all the time. Let’s find the value of the function when x=2:
(2-2)/|2-2| = 0/0 = undefined.
Okay. This function is undefined when x=2. This means there is a hole, or an asymptote, or a break or a jump or some disruption in the continuity of the function.
So, let’s talk about the limit of the function as x approaches 2:
To find the limit as x approaches 2, we need to make sure the left-handed limit and the right-handed limit both exist and are equal.
I’m going to start with the right-handed limit (remember, this means we are getting closer and closer to 2 from the top-side). We are going to do this by clever analysis:
When x=3, we get (3-2)/|3-2| = 1
When x=2.5, we get (2.5-2)/|2.5-2| = 1
When x=2.01, we get (2.01-2)/|2.01-2| = 1
See the pattern here?
When x>2, the function ALWAYS equals 1 (tricky function huh?).
So this tells us about the right-handed limit at 2:
limx→2+ (x-2)/|x-2| = 1
Now, let’s trying the left-handed limit at 2, again by analysis:
When x=1, we get (1-2)/|1-2| = -1
When x=1.5, we get (1.5-2)/|1.5-2| = -1
When x=1.99, we get (1.99-2)/|1.99-2| = -1
Again, a pattern: When x<2, the function ALWAYS equals -1.
This tells us about the left-handed limit at 2:
limx→2- (x-2)/|x-2| = -1
Since the right-handed limit at 2 ≠ the left-handed limit at 2, “the limit at 2” does not exist. The right-handed limit must equal the left-handed limit to have a “complete limit” so to speak. The concept: If I walk the path from the left, and you walk the path from the right, we better think we are going to the same place.
Here is a picture of the function just for fun. See what is happening as x approaches 2 from the right? See what is happening as x approaches 2 from the left? And, see what happens exactly at 2?