I dont understand the idea and proof behind Squeeze Theorem

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- Jul 13th 2008, 01:18 PM #1

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- Jul 13th 2008, 01:46 PM #2
Hmm, now let me recall the lecture I had myself

Lets say we want to know what the limit $\displaystyle \lim_{x \rightarrow 0}$ approaches for the function $\displaystyle x \ \sin(x)$

Now we know $\displaystyle \sin(x)$ oscillates between $\displaystyle 1$ and $\displaystyle -1$

So we can say:

$\displaystyle -1 < \sin(x) < 1$ (We know the value of the limit must be somewhere between these two values.)

But the original question asks for $\displaystyle \lim_{x \rightarrow 0} x \ \sin(x)$

So multiply throughout the equation with $\displaystyle x$

$\displaystyle -x < x \ \sin(x) < x$

We are given that $\displaystyle x$ approaches $\displaystyle 0$

$\displaystyle 0\leq \lim_{x\to0}( x \sin x )\leq 0$

Now observe, the value must be 0. It lies between 0 and... well uhm... 0 So it must be 0 itself.

And so we have calculated the limit!

**(The proof behind it will be in your textbook, go through it again now that I've shown you the idea behind the theorem.)**

- Jul 13th 2008, 02:07 PM #3

- Jul 13th 2008, 02:10 PM #4