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About Alon Mandes
Expertise
Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience
1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

You are here: Experts > Teens > Homework/Study Tips > Calculus > sqeeze theorem

Calculus - sqeeze theorem

Expert: Alon Mandes - 7/23/2008

Question
how can i prove this squeeze theorem?

lim             sin t
x approaches 0 ______=1
                 t

Answer
we will prove it in 3 different ways:
1) Lopital's low
Lim [f(x)/g(x)] = Lim [f’(x)/g’(x)]
Lim sin(t)/t= Lim cos(t)/1 = 1 when t = 0

2) Tlaylor series
The function sin(t) can be written as
t- t^3/6+t^5/120+....
So sin(t)/t =1-t^2/6+t^4/120+...
Lim (t-->0) = 1+0+0+…=1

3) For t very small, the form sin(t)< t< tang(t)
Is very true. The form can be written as
Sin(t)<t & t< sin(t)/cos(t).
Sin (t)< t --> sin(t) /t<1
t<sin(t)/cos(t)-->sin(t)/t>cos(t)
So now we have the law of ‘sandwich’:
Lim cost = 1 & Lim 1=1 then Lim sin(t)/t = 1

Remarks :
1.   The form sin(t)< t< tang(t) can be proved by drawing a unit circle where x is the angle.
2.   ‘Sandwich’ rule : If g(x)<f(x)<h(x) & Lim g(x)=Lim h(x)=L then Lim f(x)=L

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