Limit at a point of discontinuity
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- Let's say that f(x) is equal to the absolute value of x minus three over x minus three
- and what I'm curious about is the limit of f(x) as x approaches three
- and just from an inspection you can see that the function is not defined when x is equal to three - you get zero over zero: it's not defined
- So to answer this question let's try to re-write the same exact function definition slightly differently
- So let's say f(x) is going to be equal to - and I'm going to think of two cases:
- I'm going to think of the case when x is greater than three
- and when x is less than three
- So when x is - I'll do this in two different colors actually
- When x - I'll do it in green - that's not green
- When x is greater than three...
- When x is greater than three, what does this function simplify to?
- Well, whatever I get up here, I'm just taking the..
- I'm going to get a positive value up here and then I'm...
- Well, if I take the absolute value it's going to be the exact same thing, so let me...
- For x is greater than three, this is going to be the exact same thing as x minus three over x minus three
- because if x is greater than three, the numerator's going to be positive, you take the absolute value of that, you're not going to change its value
- so you get this right over here or, if we were to re-write it...
- ...if we were to re-write it, this is equal to, for x is greater than three, you're going to have f(x) is equal to one
- for x is greater than three
- Similarly, let's think about what happens when x is less than three
- When x is less than three, well, x minus three is going to be a negative number
- When you take the absolute value of that, you're essentially negating it
- so it's going to be the negative of x minus three over x minus three
- or if you were to simplify these two things, for any value as long as x doesn't equal three
- this part right over her simplifies to one, so you are left with a negative one
- negative one for x is less than three
- I encourage you, if you don't believe what I just said, try it out with some numbers
- Try out some numbers:
- 3.1, 3.001, 3.5, 4, 7
- Any number greater than three, you're going to get one
- You're going to get the same thing divided by the same thing
- and try values for x less than three:
- you're going to get negative one no matter what you try
- So let's visualise this function now
- So, now you draw some axes...
- That's my x- axis
- and then this is my...
- This is my f(x) axis - y is equal to f(x)
- and what we care about is x is equal to three
- so x is equal to one, two, three, four, five
- and we could keep going...
- and let's say this is positive one, two, so that's y is equal to one
- this is y is equal to negative one and negative two
- and we can keep going...
- So this way that we have re-written the function
- is the exact same function as this
- we've just written [it] in a different way
- and so what we're saying is...
- is we're...
- Our function is undefined at three
- but if our x is greater than three, our function is equal to one
- so if our x is greater than three, our function is equal to one
- so it looks like...
- It looks like that, and it's undefined at three
- and if x is less than three our function is equal to negative one
- so it looks like - I'll be doing that same color
- It looks like this...
- It looks like...
- Looks like this...
- Once again, it's undefined at three
- So it looks like that
- So now let's try to answer our question:
- What is the limit as x approaches three?
- Well, let's think about the limit as x approaches three
- from the negative direction, from values less than three
- So let's think about first the limit...
- ...the limit, as x approaches three...
- ...as x approaches three, the limit of f(x)...
- ...as x approaches three from the negative direction
- and all this notation here - I wrote this negative as a superscript right after the three - says
- Let's think about the limit as we're approaching...
- ...let me make this clear...
- Let's think about the limit as we're approaching from the left
- So in this case, if we get closer...
- If we get...
- If we start with values lower than three
- as we get closer and closer and closer...
- So, say we start at zero, f(x) is equal to negative one
- We go to one, f(x) is equal to negative one
- We go to two, f(x) is equal to negative one
- If you go to 2.999999, f(x) is equal to negative one
- So it looks like it is approaching negative one if you approach..
- ...if you approach from the left-hand side
- Now let's think about the limit...
- ...the limit of f(x)...
- ...the limit of f(x) as x approaches three from the positive direction, from values greater than three
- So here we see, when x is equal to five, f(x) is equal to one
- When x is equal to four, f(x) is equal to one
- When x is equal to 3.0000001, f(x) is equal to one
- So it seems to be approaching...
- It seems to be approaching positive one
- So now we have something strange
- We seem to be approaching a different value when we approach from the left
- than when we approach from the right
- and if we are approaching two different values then the limit does not exist
- So this limit right over here does not exist
- or another way of saying it:
- The limit...
- ...the limit of...
- (Let me write this in a new color - I have a little idea here)
- ...the limit of a function f(x) as x approaches some value c is equal to L if and only if...
- ...if and only if the limit of f(x) as x approaches c from the negative direction is equal to the limit
- of f(x) as x approaches c from the positive direction which is equal to L
- This did not happen here -
- the limit when we approached the left was negative one,
- the limit when we approached from the right was positive one,
- So we did not get the same limits when we approached from either side
- So the limit does not exist in this case
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