In first semester calculus, students often find limits confusing because when you substitute the value x is supposed to approach, you get nonsense like 0/0 or infinity. There is, however, an approach that can get calculus limits simplified into expressions where all you have to do is plug in the value of x given to get your answer. In beginning calculus this usually involves some sort of simple algebraic manipulation.

Let’s consider an example. Look at:

Notice that if we simply set x =1 we are in trouble, because that just gives us 0/0. So ask yourself, is there a way to rewrite the numerator and/or denominator so that this limit can be easily calculated? There is. First notice that:

We can also write:

This is great because now we can express the limit in the following way:

Now we’ve got something that’s easy to evaluate. Just set x = 1 and we have the answer:

As you get into more advanced calculus, you will see that a technique called L’Hopital’s rule allows you to handle limits like this in a more sophisticated way. But for now, isn’t it interesting that you have to use your algebraic skills to do calculus?