The other day I was browsing my old calculus book and came across a limit often presented early in textbooks, sin(x) divided by x. Here is the limit:

Well its pretty easy to see this is a problematic expression. As x goes to 0 we have 0/0. If you’re just learning basic calculus, you’re probably at a loss as to what to do, because there isn’t really any algebraic manipulation that can be done to simplify the limit and get a quick answer.

So I’m reading in my calculus book about this and what do they do? Well they have 2 or 3 pages of convoluted discussion sure to give any math student a major headache. I read this stuff and think what on earth is the author thinking?

It turns out there are two approaches that make the computation of this limit easy. The first is to apply L’Hopitals rule. In this case we’ve got an improper form 0/0 so can take derivatives to get the answer in a flash:

Here’s another approach. Expand sin(x) in a Taylor series:

Now we can write:

From this, its clear that:

This example shows that having more information about mathematics than the teachers are willing to present to you, generally  makes things easier to compute. If it were up to me I would leave the example out of the textbook until you’ve got the tools to tackle it.

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