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QUESTION: what is the lim as X approaches 0 ln(1-x)-sinx/x
ANSWER: The lim of our function is composed of 2 forms : ln(1-x) & sinx/x.
Now, well known that Lim x->0 {sinx/x} = 1 . I already proved it
former question. You may brows through my answered question & you
will find it, proved in 3 ways. Now in the 1st part x=0 is not a
singular point for ln(1-x), in fact its continuous & ln(1)=0. We
can thought prove it by the definition of the Lim :
Wee need to prove that for every small ξ>0 exists small η>0 such
that : for all x with |x|<η we have |ln(x)|<ξ .
We agree that |ln(x)|<|x| so we can claim that if |lnx|<ξ then
|x|<ξ . Hence, if we choose η=2ξ, we will gain |x|<ξ --> |x|<2ξ -->
|x|<η . Q.E.D
Alon.
---------- FOLLOW-UP ----------
QUESTION: Let R be the Region bounded by the y axis and the graphs y= x3/(1+x2) and y= 4-2x. Find the area of R.
The area of R is the finite integral of (4-2x)-[x^3/(1+x^2)]. from
zero to 1.49. The two graphs intersect in x=1.49.
So , you need to calculate ∫4-2x-x^3/(1+x^2) dx {x->0:1.49}.
Now, to calculate ∫x^3/(1+x^2) dx , use the substitute method :
Let u=1+x^2 -> 2xdx=du -> dx=du/2(u-1)^(1/2). So the integral will become : ∫(u-1)2u du. & this integral is easy to perform.
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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