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Squeeze Theorem Limit

462chevelle

Gold Member
304
9
1. The problem statement, all variables and given/known data

Lim (cos^2(t))/(t^2+1)
t->∞

2. Relevant equations
squeeze theorem -1<=Cosx<=1



3. The attempt at a solution
I have
-1<=Cos(t)<=1
(-1)^2<=Cos^2(t)<=(1)^2
(1)/(t^2+1)<=(Cos^2(t))/(t^2+1)<=(1)/(t^2+1)
I took both of limits of the 2 outsides as t->0
i got -1 and 1. so the limit should not exist. But i think this is incorrect. Any hints on what im doing wrong?
 

462chevelle

Gold Member
304
9
no wonder. am i supposed to take both sides limits at infinity instead of zero?
 

HallsofIvy

Science Advisor
Homework Helper
41,644
839
Uhh, yes, the problem says "[itex]t\to \infty[/itex]", not 0. But you don't want to say "[itex]-1\le cos(t)\le 1[/itex] therefore [itex]1\le cos(t)\le 1[/itex]"! If x is somewhere between -1 and 1 then [itex]x^2[/itex] is somewhere between 0 and 1, NOT "between 1 and 1"! Draw a graph of [itex]y= x^2[/itex] for [itex]-1\le x\le 1[/itex] to see that.
 

462chevelle

Gold Member
304
9
oh, ok. so when i put cos like
-1<=Cos(t)<=1 i should start out like
-1<=cos^2(t)<=1.
that way i dont have to square the cos then get from 1<cos<1

so there i get the limit of each is 0 so the limit of the entire function must be 0
 

HallsofIvy

Science Advisor
Homework Helper
41,644
839
Better is 0<= cos^2(t)<= 1 like I said.
 

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