Advanced Limit Evaluation Examples

Evaluate .

To start off, substitute 4 in for x. Maybe we'll get lucky and solve it immediately?

Nope. Shoulda seen that one coming. Let's factor instead, to try and eliminate the zero in the denominator.

Since (x – 4) exists in both the numerator and denominator, we can divide those factors out. Direct substitution is now clear to go through without a hitch.

That was completely hitchless, just how we like it.

Evaluate .

Step 1 is to try direct substitution.

Surprise, it didn't work. So, step 2 is panic. Step 3 is to try and factor and cancel terms. Unfortunately, the numerator doesn't look very friendly. Step 4 is crying. A lot. Then we head to step 5, polynomial division.

By dividing the numerator by the denominator, we get:

Great, it worked. That means we can go to the final step, substituting in -3 and finding the limit.

= (-3)² – 4

= 9 – 4 = 5

Evaluate .

Same first step as usual, plug and chug—and get . Great.

We obviously can't factor and polynomial division isn't going to cut it. We have a radical in the numerator, so the multiplying the top and bottom by the conjugate is our next bet.

Simplifying the numerator has wrapped things up nicely for us. It must be because we've been such good Shmoopers this year. It's good math and good karma.

Directly substitution of 2 for x was all we had left to do, so we did it.

Evaluate .

It's apparent pretty quickly that we have  if we try direct substitution. The answer, then, is obvious: don't try direct substitution. Instead, let's plug in some points close to 3 from each side and see what happens.

We get increasingly smaller (or maybe decreasingly big) numbers coming from the left, and increasingly big numbers from the right. The limit is approaching -∞ from the left and +∞ from the right. They disagree, so the limit as a whole Does Not Exist.