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Indeterminate Forms and L'Hopital's Rule

  1. Jan 21, 2013 #1
    1. The problem statement, all variables and given/known data

    lim ln(x-1)/(x2-x-4)
    x->2

    2. Relevant equations



    3. The attempt at a solution

    Well, I thought that every time I had answers as 0/0, 2/0 or 0/2, for instance, they would constitute as indeterminate forms.

    I have the answer sheet for this problem. It says "answer: 0/-2 = 0".
    Why isn't it considerate indeterminate form?

    My prof. didnt use L'Hopital's Rule here but direct substitution.
     
  2. jcsd
  3. Jan 21, 2013 #2

    Mark44

    Staff: Mentor

    The only indeterminate form in your list is 0/0.

    2/0 is undefined (which is different), and 0/2 = 0.
    L'Hopital's Rule does not apply in this problem. It can be used on the indeterminate forms [0/0] or [∞/∞].
     
  4. Jan 21, 2013 #3
    So indeterminate form = 0/0
    undefined = 2/0

    And I apply L'Hopital's rule for both indeterminate and undefined answers?

    Because I had solved some problems resulting in 6/0 using L'Hopital's rule.

    0 = 0/2

    Is that correct?

    obs: having these "small" things clarified make ALL the difference. Thanks a lot! This forum is crucial in my studies. I have an exam approaching and need to have all these info clear in my mind.
     
    Last edited: Jan 21, 2013
  5. Jan 21, 2013 #4

    Mark44

    Staff: Mentor

    NO - just indeterminate forms [0/0] or [∞/∞].
    L'Hopital's Rule can result in a value that is undefined, but you can't use it on an expression that is undefined. Do you understand the difference in what I wrote?
    Yes. 0 divided by anything other than 0 is 0.
     
  6. Jan 21, 2013 #5
    I'd like to get three examples to make sure I understood, if you don't mind. Here it is:

    EXAMPLE 1
    Lim [1/x - 1/(x2)]
    x-> 0

    Direct substitution will give me -1/0

    Should I apply L'Hopital's Rule ?

    If so, I'll have 0/2.
    The book says the answer is ∞

    EXAMPLE 2

    Lim x2 + 2x + 3/(x-1)
    x->∞

    I applied L'Hopital's Rule here twice and got 2/0.

    My book says the answer should be ∞

    EXAMPLE 3

    Lim 3x2 - 2x + 1/(2x2 + 3)
    x->∞

    Applying L'Hopital's rule twice will give me 3/2. And that's the same answer the book gives me. So I guess that's right?
     
  7. Jan 21, 2013 #6

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    No, because it's not of indeterminate form. It's no 0/0 .

    B.T.W: The result is -∞ not ∞ .
    If you mean
    Lim (x2 + 2x + 3)/(x-1) ,
    x->∞

    then it is indeterminate, of the form ∞/∞ .

    After applying L'Hôpital's rule once it will be of the form ∞/1, thus not indeterminate.
     
  8. Jan 21, 2013 #7

    Mark44

    Staff: Mentor

    How do you figure that? Direct substitution gives 1/0 - 1/0, both of which are undefined.
    No. L'Hopital's Rule applies only to limits of the form f(x)/g(x). It doesn't apply to sums, differences, or products.
    Is the limit expression (x2 + 2x + 3)/(x - 1)? What you wrote is x2 + 2x + (3/(x - 1)).

    Apply L'H once to get your limit.
    Yes, that is correct, but you need parentheses around the numerator of your limit expression, like this:
    Lim (3x2 - 2x + 1)/(2x2 + 3)
    x->∞
     
  9. Jan 21, 2013 #8
    =D Thank you both so much for your clear explanations. That will prevent me from getting wrong answers from now on. I'll pay attention to these rules when applying L'Hopital's
     
  10. Jan 22, 2013 #9
    Just a question to sum it all up:

    Based on what I learned here, L'hopital's rule does not apply to sum, differences and subtractions. It also only applies to answers in the indeterminate form of 0/0 and infinity/infinity. However, you said that 1/0 is undefined. So if I am supposed to apply l'hopitals rule to a problem and direct substitution results in 1/0, should I write the answer as undefined?
     
  11. Jan 22, 2013 #10

    Mark44

    Staff: Mentor

    If you apply L'Hopital's Rule and end up with something of the form 1/0, the answer could be ∞, -∞, or undefined.

    Some examples:
    $$\lim_{x \to 0} \frac{1}{x^2} = ∞$$
    $$\lim_{x \to 0^+} \frac{1}{x} = ∞$$
    $$\lim_{x \to 0^-} \frac{1}{x} = -∞$$
    $$\lim_{x \to 0} \frac{1}{x} ~~ \text{is undefined}$$
     
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