Limit Theorems

Khan Vid (This lesson also offers practice quizzes)

So you might recall limits… they’re pretty easy right, well get prepared because we’re gonna take this to another level (For me it was a bit intimidating at first but once you get it, you get it.)

I hope you remember how to find the limit using equations/graphs because we’re gonna need that right now. 

For the sake of keeping everything in a proper format, let’s just say that for every equation I put we are looking for the limit as we approach c. 

In the video, Sal establishes that lim(f(x))xc=L and lim(g(x))xc=M. Don’t worry about the specific value of L or M, unless it really trips you up, just think of them as any real number.

Now that we’ve established this, let’s dive into the specific properties.

Sum/Difference Rule

The sum rule is a simple one, here’s the equation

lim(f(x)+g(x))=lim(f(x))+lim(g(x))= L+M

As you can see, when we’re adding two functions inside of the limit, we can just add the separate limits of those functions.

The difference rule is the opposite of this:

lim(f(x)-g(x))=lim(f(x))-lim(g(x))= L-M

Product Rule

lim f(x)*g(x)=lim f(x)*lim g(x)= L*M

When two functions are being multiplied, we multiply their individual limits.

Quotient Rule

lim f(x)/g(x)= lim f(x)/ lim g(x) = L/M

When two functions are being divided, we divide their individual limits.

Power Rule

lim [f(x)]n= [lim f(x)]n=Ln

Constant Multiple Rule

lim [a*f(x)]= a*lim f(x) = a*L

Root Rule

Limnf(x)=nlim f(x)

A Misunderstanding I Had that You Might Have Too

Look at the following practice problem (on Khan.)

First thing you might’ve noticed is that the limits of f(x) and h(x) as they approach -1 don’t exist. But keep this in mind we are NOT looking for the individual general limits of f(x) or h(x), we are looking for the general limit of both of these combined.

But before I dive further, let’s take a step back into the basics. What needs to be true for a general limit to exist? The function needs to approach the same value from both sides. Soooo let’s get to it.

Just like any other function, we must find the limit from the left and right of both of these functions combined. Because they’re being subtracted, we’re going to use the difference property.

Let’s start with the left. As x approaches -1 from the left, the lim f(x)=2 and lim h(x)=1. Now we can proceed to put the difference rule to work: 2-1=1. So now we know lim f(x)-g(x)=2-1=1 as x approaches -1 from the left.

Now let’s work on the right. As x approaches -1 from the right lim f(x)=1 and lim h(x)=0. If we put the difference rule to work once again we get 1-0=1.

From both sides the limit is 1, so even though the general limit doesn’t exist for the individual functions, we can confirm that there is a general limit for both functions combined, which is 1.

If you want to practice this yourself, do this quiz on this khan lesson titled “Limits of Combined Functions: sums and differences.” 

Remember that you can email me (or text me if you have my phone number) for any questions regarding this question or anything on these notes.