# Thread: Using the Squeeze Theorem on Cosine Function

1. ## Using the Squeeze Theorem on Cosine Function

I'm trying to understand the proof of this function:

$
\displaystyle \lim_{x\to 0 } x^2cos 20 \pi x =0}
$

Here is the part that I understand:

Let $f(x)=-x^2, g(x)=x^2cos 20 \pi x, h(x)=x^2$
Then, $-1 \leq cos 20 \pi x \leq 1$

But then, they go on to state this, which I can't figure out:

$-x^2 \leq x^2cos 20 \pi x \leq x^2$

What I don't understand is how the x-squared terms can always be either less than -1 or greater than 1. I understand that the middle term oscillates between -1 and 1, but how do we know what x-squared is?

Can anybody help me understand this?

Thanks.

2. Originally Posted by joatmon
I'm trying to understand the proof of this function:

$
\displaystyle \lim_{x\to 0 } x^2cos 20 \pi x =0}
$

Here is the part that I understand:

Let $f(x)=-x^2, g(x)=x^2cos 20 \pi x, h(x)=x^2$
Then, $-1 \leq cos 20 \pi x \leq 1$

But then, they go on to state this, which I can't figure out:

$-x^2 \leq x^2cos 20 \pi x \leq x^2$

What I don't understand is how the x-squared terms can always be either less than -1 or greater than 1. I understand that the middle term oscillates between -1 and 1, but how do we know what x-squared is?

Can anybody help me understand this?

Thanks.
$\displaystyle\lim_{x\to 0} -x^2=\lim_{x\to 0}x^2=0$

By the squeeze theorem, the overall limit must be 0.

Did that help?

3. I did not understand what is your problem.
However, the second inequality formed by multiplying the first one by $\displaystyle x^2$.

4. Here is the graph of the squeeze theorem in action.

5. Originally Posted by joatmon
I'm trying to understand the proof of this function:

$
\displaystyle \lim_{x\to 0 } x^2cos 20 \pi x =0}
$

Here is the part that I understand:

Let $f(x)=-x^2, g(x)=x^2cos 20 \pi x, h(x)=x^2$
Then, $-1 \leq cos 20 \pi x \leq 1$

But then, they go on to state this, which I can't figure out:

$-x^2 \leq x^2cos 20 \pi x \leq x^2$

What I don't understand is how the x-squared terms can always be either less than -1 or greater than 1. I understand that the middle term oscillates between -1 and 1, but how do we know what x-squared is?

Can anybody help me understand this?

Thanks.
There is nothing there that says that "the x-squared terms can always be either less than -1 or greater than 1." Where did you get that idea? $-x^2\le x^2cos(20\pi x)\le x^2$ says nothing about how large $x^2$ is.

The point is to take the limit as x goes to 0. As x goes to 0, so does $x^2$.

6. Thanks for the help!