1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Limit Question

  1. Jan 17, 2010 #1
    1. The problem statement, all variables and given/known data
    If [tex]\lim_{x\rightarrow c} f(x) = f(c)[/tex] for all values of c, 0<or= c <or=5, and f(0)[tex]\neq[/tex]f(5), which of the following could be false?
    A. f(4) exists

    B. f'(1) exists

    C. [tex]\lim_{x\rightarrow2^+} f(x)[/tex] exists

    D. [tex]\lim_{x\rightarrow3} f(x) = \lim_{x\rightarrow3^+} f(x)[/tex]

    E. [tex]\lim_{x\rightarrow0^+} f(x) \neq \lim_{x\rightarrow5} f(x)[/tex]

    3. The attempt at a solution
    I tried process of elimination, but they all seemed true.
    A. f(4) exists simply by the first condition.
    B. f'(1) should exist because the limit for f(1) exists.
    C. this should exist if limit of x to 2 f(x) exists.
    D. If the limit on both sides exist, it should be equal to the right side or the left side as well.
    E. limit of 0 from the right side = limit of 0 from both sides, which is = f(0); the other part is = f(5), so this statement should be true.

    Clearly I've made a mistake in my logic somewhere.. :(

    EDIT: Oh, wait, could it be B? If you do a cusp at x=1, then the f(1) exists, but f'(1) doesn't.
  2. jcsd
  3. Jan 17, 2010 #2
    It looks to me B. Because, eventhough f, with the given properties/conditions, is continuous at 1, remember that continuity does not imply differentiability(while the converse is true).
  4. Jan 18, 2010 #3

    You got it right! For instance, [tex]f(x)={\frac {\sqrt { \left| x \right| }}{{x}^{2}+1}[/tex] vanishes at x=0; while its derivative [tex]1/2\,{\frac { \left| 1 \right| }{\sqrt { \left| x \right| } \left( {x}^{2}+1 \right) }}-2\,{\frac {\sqrt { \left| x \right| }x}{ \left( {x}^{2}+1 \right) ^{2}}}[/tex] is not defined at the same point.

  5. Jan 18, 2010 #4
    Thanks for the confirmation guys.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Limit Question Date
Quick Limits Question Dec 22, 2017
Question about finding the limit of a^(1/n) Nov 1, 2017
Limit question Nov 7, 2016
Limit question May 21, 2016
Sequence Convergence/Divergence Question Apr 3, 2016