## Chapters

# Applications

Remember in the beginning, we said that when we get to derivatives, we'll be doing limits all the time? Now is the time to learn some useful limits that you are going to rely heavily on in the future. The first one of these is

The solution to this limit is not immediately obvious, so we are going to take a detour and review something that you (should have) learned in Precalculus.

In the graph above, we have 3 values :

If you don't immediately have an answer, think for a few minutes about how we could order these three values in an inequality. One possible (wrong) answer is

Once you think you have an answer, continue on.

Looking at the graph tells us that

Now, we can use this property to evaluate our limit! Dividing by $\sin{\theta}$ gives

Oops! We have our value, but it's upside down! Let's plug in $x$ to this anyways, and see what we can do. Well, we can just take the reciprocal of all of them and we'll be good to go, right?? No [Why not?]. However, we *can* take the reciprocal, and then reverse the signs. So now, we have

We know for a fact that

So, now we say for sure that

This is pretty redundant, and implies that

Yes, really - it's just 1. Now, try a two-step limit. Don't forget to use what you just learned! 2.

This is one example of how the limit we just learned can sneak up on you! Here's how we transform it into two problems we can solve:

One times one is... one? Yep.