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Question
prove that the fuction f(x)=x^3-2x+3 is countinous..
Let's follow the ξ-δ proof : Lim f(x) Where x->c means :
For every ξ>0, there exists a δ>0 such that whenever |x-c|<δ,
|f(x)-f(c)|<ξ.
|f(x)-f(c)|=|x^3-2x+3-c^3+2c -3|=|x^3-c^3-2x+2c|=|x^3-c^3+2(x-c)|
So, we know that |x^3-c^3+2(x-c)|<ξ.But |x^3-c^3|=|x-c||x^2+cx+c^2|
So, |[x-c][x^2+cx+c^2]+2[x-c]|<ξ
|[x-c]||2+[x^2+cx+c^2]|<ξ
|x-c||2+M(x,c)|<ξ
M(x,c) has upper bound μ in the interval [x-δ,x+δ], so
|x-c||2+μ|<ξ
|x-c|<ξ/|2+μ|, So δ is chosen to be ξ/|2+μ| .
All you have to do is decide how large μ has to be.("To do so, use
the function M(x,c) in the interval [x-δ,x+δ] to evaluate the upper
bound").
Alon.
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Answers by Expert:
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".
Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.
Education/Credentials
M.A in Mathematics & Bs.c in Electronics.
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