1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Prove lim f(x) as x approaches a is 0?

  1. Sep 21, 2011 #1
    1. The problem statement, all variables and given/known data

    It is given that the limit as x->a of ( f(x)/(x-a) ) is 3. Prove using the espilon-delta theorem of limit that the limit as x->a of f(x) is 0.

    3. The attempt at a solution

    so it is known that: |( f(x)/(x-a) ) - 3| < E1 when |x-a| < d1

    |( f(x)/(x-a) ) - 3| ≤ |f(x)/(x-a)|+3 < E1

    |f(x)| < (E1-3)|x-a|

    (E1-3)|x-a| ≤ C(E-3)

    |x-a| ≤ 1=C

    |f(x)| < (E1-3) when d1=min{ 1, (E-3) }

    and we have to prove that |f(x)-0| < E when |x-a| < d

    so can i just say that:

    |f(x)-0| < E2
    |f(x)| < E2 when |x-a| < d2

    So E2=E1-3 and d2=d1=min{ 1, (E-3) }

    So |f(x)-0| < E1-3 when |x-a| < d1

    is this right? i am not sure if i am making any logical sense in equating the two epsilons?

    my intuition tells me that f(x) is zero at "a" because for f(x)/x-a to have a limit at "a", f(x) must be a factor of (x-a) so as to cancel out with the denominator. so if f(x)=(x-a)(p(x)), then f(a)=0, but i am not sure how to prove that using epsilon-delta, and i am not even sure if f(x) HAS to be a factor of (x-a)...sinx/x after all has a limit at 0 but sinx doesnt cancel with x.

    help? :D
  2. jcsd
  3. Sep 21, 2011 #2


    User Avatar
    Science Advisor

    The denominator of f(x)/(x- 3) goes to 0 as x goes to 3. If f(x) goes to anything other than 0, then the limit of f(x)/(x-3), as x goes to 3, could not be finite.

    It does NOT follow that f(3)= 0 because you are not told that f is continuous there. But that is not asked.
  4. Sep 21, 2011 #3
    why would it not be finite if f(x) did not go to 0?

    specifically i want to know if my proof is right? my professor wants it to be proved using epsilon-delta, not just words..
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook