September is around the corner and that means calculus students the world over are getting ready to tackle limits! For many students, computing limits is their first introduction to the more grown up world of calculus, as compared to the way they’ve been doing math so far. Limits seem mysterious and you may even think there is no way you’ll pass calculus.

But I’m here to tell you that limits are actually a very simple idea. Before you get bogged down into really understanding what they’re about and all those epsilon-delta proofs, chances are you just want to be able to calculate some. After all unless you’re a math major that is 95% of what you need to do in order to get through calculus. That’s the good news. Because in reality, doing limits just involves doing a little trick here and there to rewrite whatever problem they’ve given you and then plug and chug some numbers. Let’s take an example.

The first limits you face are probably going to involve some kind of quotient, like:

What happens if you just plug in x =2? Well you get 0/0, which at this point in your calculus career is a meaningless statement. So what is the trick to doing this limit? If you’ve passed algebra you’ll be pleasantly surprised: all we have to do is factor the numerator. Notice:

When you rewrite the numerator this way, the limit becomes:

Now we can cancel the term in the denominator, leaving a simple expression where we can plug in x = 2 to compute the limit:

That’s all there is to it. Nothing mysterious, and not a word about epsilon or delta or approaching this or approaching that. Next time, we’ll take a look at some more limits that look hard to compute at first glance but are really quite simple.

For more examples of calculus homework, check out my ebook Calculus without Limits.