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Calculus/limit theorems

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Expert: Alon Mandes - 10/5/2009


Question
Hi.

I have been wondering of why the the limit of a polynomial as x approaches something we can distribute the limit
ie: limit x -> a  (x^5+3x^2+6x+8) = a^5 +3a^2+6a+8

While the limit as x -> a of 1/x^2 +1/x +x^2
= limit x -> a 1/x^2 +limit x -> a 1/x +limit x -> a x^2
= 1/a^2 +1/a +a^2

What is the difference between the limit of a polynomial and the limit of an algebraic with x as the denominator ?


Answer
Hello Hamad,
The limit definition doesn't change, whether it's a polynom or a fraction. But the issue of
continuesity does. For example : (x²-9)/(x-3) has singularity at x=3. We can't plug 3 in the
denominator. The trick here is to perform a mathematical trick : (x²-9)/(x-3)=(x+3)(x-3)/(x-3).
Then we get x+3. This is polynom %26 we can plug 3 %26 get 9 .
But when using The Eplison-Delta definition, there won't be a great deal of importance whether
the function is continues or not. The definition covers it all.
It's pretty common that polynoms are continues over all x , so in this case it's fine to plug
the limit, while denominators are not.
Hope I made it much clear .

Alon.

Calculus

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Alon Mandes

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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 .

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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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M.A in Mathematics & Bs.c in Electronics.

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