# Thread: "Squeeze/Sandwich Theorem" I'm lost

1. ## "Squeeze/Sandwich Theorem" I'm lost

Hi, is it just me or do text books don't explain the Squeeze/Sandwich theorem properly? I'm always unsure of the domain required. Some help would be very much appreciated =]

2. The domain isn't really much of a problem, so long as the two bounding functions converge there. The sandwich principle says

If a function $f(z)$ is bound between two convergent functions $g(z),h(z)$ that converge to $l_g,l_h$ respectively then $l_f$ is somewhere between $l_g$ and $l_h$ (providing that $f(z)$ is a convergent function). Hence, if $l_g=l_h$ then $l_f=l_g=l_h$

Please observe that, if a function is bound between two functions that do not converge to the same limit, this does not mean that the first function converges.
Take, for example $f(x)=\sin(x)$ and $g(x)=\frac{3x^2+3x}{(2x+1)(x+4)},h(x)=\frac{(1-x)(2+2x^2)}{x^3}$. Clearly $\lim_{n\rightarrow\infty}g(x)=3/2$, $\lim_{n\rightarrow\infty}h(x)=-2$ and $l_h\leq-1\leq\sin(x)\leq 1 \leq l_g$.
Just because $f(x)$ is bound between two convergent series does not mean that it converges

Paul

3. hrmm...i think i wasn't clearly pointing out the problem . Let's say we try to prove:

lim x-> 0 x.cos(1/x) = 0

squeeze theorem states that: f(x) <= g(x) <= h(x)

and
limf(x) = lim(hx) = L
then limg(x) = L

i don't know where to begin:

i know you must find a domain for which x.cos(1/x) lies in and then take the limits of the left bound and right bound, which would equal to the limit of the function being squeezed.

But, how to find the domains? Thanks for the reply earlier Paul.

4. Originally Posted by tasukete
lim x-> 0 x.cos(1/x) = 0

squeeze theorem states that: f(x) <= g(x) <= h(x)

and
limf(x) = lim(hx) = L
then limg(x) = L

Let's first work with $\cos \frac{1}{x}$

The cos graph will be oscillating between [-1 ; 1]

Therefore:

$-1 \leq \cos \frac{1}{x} \leq 1$

Multiply by x through the inequality.

$-x \leq x \cos \frac{1}{x} \leq x$

If x approaches zero, we obtain the following:

$0 \leq x \cos \frac{1}{x} \leq 0$

And if the left and right is zero, then the middle must also be zero... (That's logical! )

5. so for these kind of problems you've got a have a mental picture of the graph? i've done a couple of problems and starting to get the feel of it now. thnx Pual =]

6. Originally Posted by tasukete
so for these kind of problems you've got a have a mental picture of the graph?
Not really. You know very well that sin and cos oscillates between [-1 ; 1]

Originally Posted by tasukete
i've done a couple of problems and starting to get the feel of it now.
Practise makes perfect.

Originally Posted by tasukete
thnx Pual =]
Not Paul!

7. Sorry. LOL pls ignore that...