Question

find the one-sided limits indicated.
f(x)={2-3x if x<2
{x-1 if x is greater than or equal to 2

a. lim f(x) approaching 2 from the left
b. lim f(x) approaching 2 from the right

Answer 1

OK. How far did you get with this? Do you know what the one sided limits are it is asking for, or is it the system that is confusing you?

Answer 2

I havent gotten anywhere with it. I dont know how to do the one sided limits. The whole thing is confusing me.

Answer 3

OK. Then let me start with the one side part. It is a pretty simple concept related to the number line.

Answer 4

If I am coming at any point from the left, I start below it on the number line. So for your problem with 2, anything BELOW 2 is the left of 2, and anything ABOVE 2 is the right of 2.

Answer 5

Does that much make sense now?

Answer 6

yes

Answer 7

OK. What the system of equations part means is, when the x value is in a specific range, you use one calculation. when it is in another range, you use the other. So lets start with the left. For the left of 2 it is any value less than 2. So when you look at the system the first part says:

2-3x if x<2

So, for the left of 2, it is the limit of 2-3x when x approaches 2.

Answer 8

What limit do you get there?
\[\lim_{x\to2^-}2-3x\]

Answer 9

-4?

Answer 10

Yes! Absolutely! And if you write an answer on paper, remember the - sign I did:
\[\lim_{x\to2^-}2-3x=-4\]\(\quad\; \uparrow\) That - right there means from the left.

Answer 11

The second part of this is the right of 2, so any number greater than 2. So they are looking at the other part of the composite function.

\(x-1\) if \(x\ge 2\) The fact that it is greater or EQUALS to does not matter. In limits, we rarely care what something equals. We want to know where it is going towards, not where it is.

So you want the limit:
\[\lim_{x\to2^+}x-1\]

Answer 12

so do i plug in the 2 for x and do 2-1?

Answer 13

Yah, linear equations are very easy to evaluate. That is why they are using them for the start of limits from the left and right.
\[\lim_{x\to2^+}x-1=1\]\(\quad\; \uparrow\) And again, the sign has meaning here. + means from the right.

Answer 14

Another thing they may ask is if the limit exists at 2. Well, for the limit to exist, the limit from the right and left must match. We got 1 and -4. Those are NOT the same, so the limit at 2 does not exist. That is one reason why limits from the right and left are important.

Answer 15

ok i think i got it. so there is nothing else left to do to this equation?

Answer 16

You have answers for (a) and (b), -4 and 1. That is all they asked. The other thing I said is what they are leading to with this.

Answer 17

o ok well that wasnt as difficult as I expected thank you

Answer 18

The big confusion is they started talking right and left at the same time they threw out a composite function. It seems like a lot. You just needed to see it broken into parts.