First let's talk about some RULES that govern limits. Like with all things in math, there's stuff you can and can't do. Here's what's valid when working with limits:
The limit of two functions being added is the same as the limits of each function taken seperately and then added together. This is like distributing the limit operation through the function.
The same thing applies with subtracting limits.
Example:
Constants can be pulled outside of the limit and the limit calculated independantly of the constant.
Example:
The limit of the product of two functions is the same as the product of the limits.
Example:
Same as above, but with division.
Example:
THEN we have to discuss INDETERMINANT FORMS. Ugh does it never end?! EVEN MORE SETUP INFORMATION:
Indeterminant forms are things you get when evaluating limits THAT TELL YOU NOTHING. If your limit ends up being in an indeterminant form, you need to rearrange it until it's not. (I'm not working off a list here like above so if I forget one of the forms, let me know and I'll add it.)
Indeterminant forms:All of these are indeterminant forms because you can't tell, for example, if the top is approaching 0 faster than the bottom is approaching 0, or if the positive infinity is getting bigger faster than the infinity being subtracted. Basically, you can't tell what's going on.
And some quick points of reference. When dealing with limits if you have a variable on the bottom and a constant on the top,
Why is that true? Well, you know how when you plug something like 1/x into your calculator, you get asymptotes at 0 where the function goes off to infinity? That's because as your denominator gets infinitely smaller (closer to 0), your function gets infinitely larger. Same thing in reverse for the other bit: as the denominator gets larger, the function gets smaller. Plug some values into your calculator and check it out if you don't believe me.
There. Now that all that's out of the way, we can talk about solving this shit.
We'll start with something easy:
"GUINDO. THAT IS NOT EASY."
Shut up, you whiner. Look at it. Plug in 2 and what do you get? 0/0, which is indeterminant, right? Well, the first thing you gotta do is figure out how to make it NOT indeterminant, and one way is by algebraic manipulation.
So let's take a closer look. We have a second degree polynomial on top and a first degree polynomial on the bottom, so your first thought should be "I wonder if this factors." If you can factor something from the top that cancels with the bottom, your indeterminant form will go away and you can solve the limit.
If you don't know how to factor, GET OUT. I'll cover that in a later post.
Awww snap, there we go, the x - 2 drops right out. Remember the footnotes on the last post where I said why you can do that even if the function you're "cancelling" is zero at the point you're using? If not, go back and read it and quit skipping ahead like a jerk.
So now it's simple enough to evaluate directly and we have our answer:
Let's evaluate something a little harder. This is one of my favourite techniques: factoring stuff that doesn't exist!
Aw man, this one is DEFINITELY infinity over infinity, AND it doesn't even factor! What are you supposed to do?!
Cheat.
Factor out the highest degree polynomial from both the top and the bottom.
"HOLY SHIT GUINDO ARE YOU FUCKING KIDDING ME THAT'S CRAZY"
Crazy enough to work!
This technique is the best. It's so simple and so easy to remember. And it works because if you distribute that x2 back through, you get the original function. There is absolutely nothing wrong with doing this, but it's so sneaky!
Simplify:
And evaluate:
IT'S SO SIMPLE. Fucking amazing, you guys. If you're ever in a bind trying to figure what the fuck you're supposed to do with a polynomial limit, TRY THAT FIRST. You don't even have to bother with figuring out your factors like in the previous example. You can even do it for stuff with ex by factoring out the highest degree of e!
Of course there's the caveat that this only really works when you're going to infinity. Fortunately most limits are either infinity or zero! And zero limits don't become a huge thing until derivatives and multi-variable calculus. AKA outside the scope of this post.
But, y'know, books like to be inconvenient, so. Just remember: algebraic manipulation should be your first resort if it's not going to infinity, and cheating should be your first resort if it is.
One more thing for this post, which is the CALCULUS part of it: L'Hospital's Rule.
It's pronounced Low-pee-tall by most professors but I like to say Luh-hospital just to shake things up.
Luh-hospital's rule says that if you're evaluating a quotient of limits, the rate of change for each function is proportional to the limit itself. Or...something like that, anyway. Basically, if you have an indeterminant form that's either 0/0 or ∞ / ∞ then you can take the derivative of the top and the derivative of the bottom and evaluate the limit of the quotient of...you know what let me just show you.
The first step evaluates to infinity over infinity. The second step takes the derivative of the top and the derivative of the bottom and compares those instead of the original functions, and that evaluates out to 0. This is possible because good ol' Luh-hospital figured out that when you're comparing the limits of two functions like this, comparing their rates of change was the same thing. You can go back and try the other examples using this method to see how it works.
Now I want you to use your imagination. Really picture this. A big flashing neon sign with sparkles everywhere and maybe a rainbow somewhere and there's definitely an angry unicorn giving you a stern glare off to the side and the sign says:
And we're done.
Next time: I never get tired of talking about limits.*
* Unless it's epsilon-delta proofs. Fuck epsilon-delta proofs.
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