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!

Simple limit question

  1. Dec 12, 2004 #1


    User Avatar

    Hello, I tried doing this limit problem:

    lim (sinx)/x^3

    here's what I did:

    (lim sinx)/(lim x^3)

    which should give something like 0,000000000...1 / 0,00000000001
    since lim sinx (x->0) = 0 and lim x^3 = 0 too. When I make the graphic in maple, the answer is +infintiy, and the answer in my book too. Could someone clarify this for me plwase
    thanks a lot
  2. jcsd
  3. Dec 12, 2004 #2


    User Avatar
    Homework Helper

    Do you know L'Hospital?

    [tex] \lim_{x \rightarrow 0} \frac{\sin x}{x^3} [/tex]

    [tex] \lim_{x \rightarrow 0} \frac{\cos x}{3x^2} = \frac{1}{0} = \infty[/tex]

    Alternatively you could use the fact that

    [tex] \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 [/tex]

    So we rewrite the first expression to

    [tex] \lim_{x \rightarrow 0} \frac{\sin x}{x} \frac{1}{x^2} [/tex]

    Applying limit laws:

    [tex] \lim_{x \rightarrow 0} \frac{\sin x}{x} \lim_{x \rightarrow 0} \frac{1}{x^2} [/tex]

    [tex] 1 \cdot \frac{1}{0} = \infty [/tex]

    Note: when you try to calculate the limit, you should try to get it without making a indeterminate form such as the one you got ([itex] \frac{0}{0} [/itex]).
    Last edited: Dec 12, 2004
  4. Dec 12, 2004 #3


    User Avatar

    Thank you very much for your help. I appreciate it :)
    What do you mean by L'hospital? In french we say L'hopital ;).. if its a theorem we haven't seen it yet.
    thanks again
  5. Dec 12, 2004 #4
    He means L'Hopital's rule. Mathworld can't seem to spell hopital :/, but read their page on L'Hopital's rule.
  6. Dec 12, 2004 #5


    User Avatar
    Homework Helper

    Sorry, my native language is spanish :smile:, and i was taught L'Hospital like that, it appears like that on our books in spanish, and in the english books too, i've seen it too as L'Hospitel, oh well :smile:
  7. Dec 12, 2004 #6


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    mad: The name of the theorem does not refer to an actual "hôpital" but as a dude whose last name was Guillaume François Antoine de L'Hospital (he was a Marquis in France I think).

    Also, be careful while using those properties of the limit such as the one Cyclovenom used, that is, since

    [tex] \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 [/tex]


    [tex]\lim_{x \rightarrow 0} \frac{1}{x^2} = +\infty[/tex]


    [tex] \lim_{x \rightarrow 0} \frac{\sin x}{x} \frac{1}{x^2} = +\infty[/tex]

    You have to watch out for the cases where the limits you "separated" have indeterminate forms (there are 7 of them). When you have a quotient, or product, or sum of limits that have this form, you cannot conclude to the value of limit. (as much as one would be tempted to conclude that [itex]\infty - \infty = 0[/itex] for exemple, we can not.)
    Last edited: Dec 12, 2004
  8. Dec 12, 2004 #7
    This page explains everything.
  9. Dec 12, 2004 #8


    User Avatar

    Thanks a lot guys!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Simple limit question
  1. Simple limits question (Replies: 4)

  2. A Limits Question (Replies: 3)

  3. Limits Question (Replies: 1)

  4. Simple limit proof (Replies: 2)