PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
When we evaluate limits that are not continuous, we can use algebra to eliminate the zero from the denominator and then evaluate the limit using substitution. When evaluating limits algebraically we can eliminate the zero in the denominator by factoring or simplifying the function.
Let's talk about evaluating limits algebraically when continuity doesn't work, here is an example how would we evaluate the limit as x approaches 2 of x squared +8x-20 over x-2. Now you can see the problem as x approaches 2 this denominator is going to 0, so division by 0 is the problem here. But remember when you're calculating limits you're answering the question what happens as we get close to 2 not what happens at 2. So let's take a look at this rational function, turns out that I can factor the numerator right. And remember you're writing a limit statements so uh this limit statement has to equal another limit statement. And you have to continue writing this limit expression until you evaluate the limits. So right now I'm just doing a little Algebra inside. This numerator can be written as a product of two linear functions right and they're both going to start with x we'll have x plus or minus something x plus or minus something. I need you to look at the factors of 20, factors of -20 that are going to give me sum of 8 and thinking about 10 and 2 they use +10 and -2 I'll get my -20 and I'll get 10x-2x which is +8 that works.
And that we see here is I've got 2 factors of x-2 now ordinarily you want to be careful about just canceling these things outright you want to think about what you're doing here but remember we can cancel these two when evaluating the limit because we don't care what happens at x=2, we care what happens at near x=2. So near x=2 these two guys are always going to be the same value and so we can cancel them. And that gives us the limit as x approaches 2 of x+10 and that limit we can evaluate using continuity right? You can just plug in 2 then you get 2+10 12. So generally when you're evaluating limits without continuity you're going to have some problem like division by 0.
You just have to look for ways to use Algebra to eliminate that problem and then evaluate the limit. Now let's take at what the the graph of this function, what does that limit equaling 12 represent? This function is not defined at 2 but it has a limit as x approaches 2 of 12, this functions graph is basically a line with a whole in it at x=2. And you could rewrite this function as y=x+10 for x not equal 2.