You are here:

Calculus/Limit Proof

Advertisement

Expert: Alon Mandes - 7/20/2009


Question

Problem #111
I am working on my summer assignment for AP Calc BC, and I am stuck on how
to prove this.  It is question #111 in the picture attached.  In case you cannot
read it, it says: "Prove that if the limit of f(x) as x approaches c is 0 and the
absolute value of g(x) is less than or equal to M for a fixed number M and all x
does not equal c, then the limit as x approaches c of f(x)g(x) equals 0."

Answer
Let's prove it by the definition of limit . Reminder :
Lim  f(x) = 0 , means :
x->c
for every δ>0 exists ε>0 , so that for evey neighborhood of x
|x-c|<δ, we gain |f(x)|<ε . Where ε depend on δ (" ε(δ) ") .
We know that g(x) is bounded by M for all x, except x=c. We asume
that at x=c g(x) is defined. ("that means g(c)≠±∞"). Let's suppose
that g(c)=N. We define Q as : Q=Max{M,N}.
We can claim the following :
0 ≤ |f(x)g(x)| ≤ |Qf(x)| for all x in the motherhood |x-c|<δ .
We can also claim that -ε ≤ |f(x)g(x)| ≤ Q|f(x)| . That means for
every x such as |x-c|<ε , we gain : -ε ≤ |f(x)g(x)| ≤ Qε .
Let's define ε'=Qε , therefore we can state that :
-ε' ≤ |f(x)g(x)| ≤ ε' , which is the same as |f(x)g(x)|<ε' for
every |x-c|<δ. Hence we proved that
Lim  f(x)g(x) = 0   
x->c

Alon.

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.

©2012 About.com, a part of The New York Times Company. All rights reserved.