Barion Pixel The limits of trigonometric functions | mathXplain

Contents of this Calculus 1 episode:

Trigonometric functions, sinx/x, sine, cosine, tangent, Continuous, Factorize, 0/0, number/0, right side limit, left side limit.

Text of slideshow

If the left-side limit equals the value of the function,

then it is left-continuous.

And now let's talk about the limits of trigonometric functions.

Here we have a few exciting cases.

Well, this is a type 0/0 limit, and we should remember that it is

Here is another splendid 0/0 case,

let's memorize this one, too.

Furthermore, these have a few mutant versions, too.

So, if we need to compute this limit:

Then we can state that

and since , it holds that , and based on the mutant version, the result is 1.

This is great, so now we should look at some exercises.

If there is 2x in the sine, but only an x in the denominator, then we have to use a trick.

First we divide* both the numerator and the denominator by x,

and then we mass-apply the previous trick.

*more scientifically speaking, we simplify by x

Well, this is a quite boring problem, but since it is already here, let's solve it.

Now comes the real thrill!

The voices tell us that we should divide by x2.

I mean simplify.

For these, it would be beneficial not to divide them by x2 individually,

but all at once.

Here is a more exciting example.

Finally, the most exciting one.

There is such a thing as:

We factor out at the bottom, too.

Multiply both the numerator and the denominator by .

And now comes a trick.

And another trick.


The limits of trigonometric functions

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