# help with squeeze theorem.

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• January 7th 2008, 10:26 PM
rcmango
help with squeeze theorem.
problem by squeeze theorem: 0<=sin^2(x)/x^4 <=1

Okay, so we know that sin^2 (x) is bounded between 0 and 1,

I believe the bottom denominator, x^4 continues on to infinity. it just keeps on going until infinity, so a number could be like 1, 1/16, 1/81 and so on ... right? thus getting closer to 0. So sin^2 (x)/x^4 = 0.

is this correct?
• January 7th 2008, 10:33 PM
Jhevon
Quote:

Originally Posted by rcmango
problem by squeeze theorem: 0<=sin^2(x)/x^4 <=1

Okay, so we know that sin^2 (x) is bounded between 0 and 1,

I believe the bottom denominator, x^4 continues on to infinity. it just keeps on going until infinity, so a number could be like 1, 1/16, 1/81 and so on ... right? thus getting closer to 0. So sin^2 (x)/x^4 = 0.

is this correct?

this is not correct, what you're trying to prove is false. say x = 1/2, then we have $\frac {\sin^2 x}{x^4} > 1$. i am thinking of x in radians here. if you want to do it in degrees, a smaller number will suffice, say x = 0.00001
• January 8th 2008, 01:46 AM
badgerigar
Quote:

problem by squeeze theorem: 0<=sin^2(x)/x^4 <=1

Okay, so we know that sin^2 (x) is bounded between 0 and 1,

I believe the bottom denominator, x^4 continues on to infinity. it just keeps on going until infinity, so a number could be like 1, 1/16, 1/81 and so on ... right? thus getting closer to 0. So sin^2 (x)/x^4 = 0.

is this correct?
Looks good to me. You might want to set it out a bit more formally though. Start with something like:
$\frac {0}{x^4} \leq \frac {\sin ^2(x)}{x^4} \leq \frac {1}{x^4}$ for all x, then take the limit for all 3 parts and conclude that the limit you are looking at is between 0 and 0, and therefore is 0.
• January 8th 2008, 10:10 AM
Jhevon
Quote:

Originally Posted by badgerigar
$\frac {0}{x^4} \leq \frac {\sin ^2(x)}{x^4} \leq \frac {1}{x^4}$ for all x

well, that makes sense. but before when we had 1 instead of 1/x^4, it didn't
• January 8th 2008, 04:38 PM
badgerigar
Quote:

well, that makes sense. but before when we had 1 instead of 1/x^4, it didn't
I know what you mean. I often don't understand people when they don't say exactly what they mean, but this time I had the advantage of just having found out that squeeze theorem is another name for the sandwich rule.
• January 8th 2008, 04:42 PM
Jhevon
Quote:

Originally Posted by badgerigar
I know what you mean. I often don't understand people when they don't say exactly what they mean, but this time I had the advantage of just having found out that squeeze theorem is another name for the sandwich rule.

hehe, i've never heard it called the sandwich rule before
• January 8th 2008, 04:59 PM
Plato
Quote:

Originally Posted by Jhevon
I've never heard it called the sandwich rule before

Do you know where the term “squeeze” comes from?
In the U.S. the national sport has been “baseball”. (I think it is now gridiron football.)
But in baseball the “squeeze play” was very important: a ‘runner’ is caught between basemen. Can you understand why most of the world does not get this reference?