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Math Help - hard limit.

  1. #1
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    hard limit.

    limf(x)
    x--+oo

    f(x)=(5^x)/(1-lnx)

    i canīt solve this. i believe the answer is +00 but i would like to see the resolution step by step.
    Sorry if i have posted in the wrong section.
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  2. #2
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by Moklin View Post
    limf(x)
    x--+oo

    f(x)=(5^x)/(1-lnx)

    i canīt solve this. i believe the answer is +00 but i would like to see the resolution step by step.
    Sorry if i have posted in the wrong section.
    it's -\infty

    use L'Hospitals rule .... and show what you get
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  3. #3
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    Yeah. Itīs -oo thanks, but we are not supposed to use LīHospitals rule to solve it. :/
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  4. #4
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    This is how I was taught to do it without L'Hospitals rule

    \lim_{x \to +\infty} f(x) = \frac{(5^x)}{(1-lnx)} = - \infty

    The trick that I used is to observe all the terms as they approach a number that is getting infinitely larger. Since it's specified we only want it as it approaches positive infinity we don't need to observe it as it approaches a number that gets infinitely smaller, since the limit from the left and the right as it approaches positive infinity will be infinitely large.

    So first I look at the top and see that 5 is raised to X. What happens to 5 as it's exponent gets infinitely larger? The result is infinitely larger, so I just substituted it with a + sign to remind myself it will stay positive. It now looks like:

    \lim_{x \to +\infty} f(x) = {\frac{+}{(1-lnx)}

    Then I observe the bottom. The limit of a constant is the constant, so we know as X gets infinitely larger 1 will stay 1.

    The interesting behavior is at ln(x). As x gets infinitely larger, it stays positive. You can test this by grabbing any calculator and taking the natural log of any extremely large positive number.

    What is 1 subtract a large positive number? A negative of course, which gives me (using my symbology) -

    \frac{+}{-}

    which comes out to be a negative. This tells me that the limit will approach negative infinity due to the behavior in the denominator.


    This trick works for simple stuff like this. Even if you won't learn L'Hospital's rule in your class until second semester calc, you should pick it up if not just to check your work.
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by sinx View Post
    This is how I was taught to do it without L'Hospitals rule

    \lim_{x \to +\infty} f(x) = \frac{(5^x)}{(1-lnx)} = - \infty

    The trick that I used is to observe all the terms as they approach a number that is getting infinitely larger. Since it's specified we only want it as it approaches positive infinity we don't need to observe it as it approaches a number that gets infinitely smaller, since the limit from the left and the right as it approaches positive infinity will be infinitely large.

    So first I look at the top and see that 5 is raised to X. What happens to 5 as it's exponent gets infinitely larger? The result is infinitely larger, so I just substituted it with a + sign to remind myself it will stay positive. It now looks like:

    \lim_{x \to +\infty} f(x) = {\frac{+}{(1-lnx)}

    Then I observe the bottom. The limit of a constant is the constant, so we know as X gets infinitely larger 1 will stay 1.

    The interesting behavior is at ln(x). As x gets infinitely larger, it stays positive. You can test this by grabbing any calculator and taking the natural log of any extremely large positive number.

    What is 1 subtract a large positive number? A negative of course, which gives me (using my symbology) -

    \frac{+}{-}

    which comes out to be a negative. This tells me that the limit will approach negative infinity due to the behavior in the denominator.


    This trick works for simple stuff like this. Even if you won't learn L'Hospital's rule in your class until second semester calc, you should pick it up if not just to check your work.
    This is good as far as it goes, but how to you handle which (if either) of the infinities are bigger than the other? In other words, you will have to prove that \displaystyle \lim_{x \to \infty} 5^x is a "larger" infinity than \displaystyle \lim_{x \to \infty} ln(x). You have not done so.

    -Dan
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  6. #6
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    Quote Originally Posted by topsquark View Post
    This is good as far as it goes, but how to you handle which (if either) of the infinities are bigger than the other? In other words, you will have to prove that \displaystyle \lim_{x \to \infty} 5^x is a "larger" infinity than \displaystyle \lim_{x \to \infty} ln(x). You have not done so.

    -Dan
    I suppose the trick in it is that regardless of which infinity is larger, you're still subtracting something infinitely large from 1 resulting in a negative, and the number in the numerator will remain infinitely positive as nothing is acting on it but X. Even though it doesn't stand for proof, you could easily say that

    \frac{5,000,000}{1-5,999,999}

    will be some negative value simply because 1 subtract any large number will be negative. However, I still talk to the professor that taught me this trick so I'll ask him to prove it for me next week and see what happens. I'll post the result. Good job pointing it out, I didn't even consider proofing it.
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  7. #7
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    Quote Originally Posted by Moklin View Post
    limf(x)
    x--+oo

    f(x)=(5^x)/(1-lnx)

    i canīt solve this. i believe the answer is +00 but i would like to see the resolution step by step.
    Sorry if i have posted in the wrong section.
    You could use the fact that

    e^x>lnx\Rightarrow\ e^x>lnx-1

    and 5>e


    \displaystyle\lim_{x \to \infty}\frac{5^x}{1-lnx}=-\lim_{x \to \infty}\frac{5^x}{lnx-1}=-\lim_{x \to \infty}\frac{\left[\frac{5}{e}\right]^xe^x}{lnx-1}


    goes to negative infinity.
    Last edited by Archie Meade; February 17th 2011 at 03:29 PM.
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  8. #8
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    My teacher solved it from me but i think he made a calculus mistake :/
    He said the answer was +oo

    Anyway...can anyone solve this?

    limx--0+ f(x)

    f(x)= xln(x)

    Without using LīHopital rule.
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  9. #9
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    Quote Originally Posted by Moklin View Post
    My teacher solved it from me but i think he made a calculus mistake :/
    He said the answer was +oo

    Anyway...can anyone solve this?

    limx--0+ f(x)

    f(x)= xln(x)

    Without using LīHopital rule.
    Again, you could utilise the fact that

    e^x>log_ex

    Hence

    xe^x>xlog_ex

    \displaystyle\lim_{x \to 0^+}xe^x=\lim_{x \to 0^+}\frac{e^x}{\left(\frac{1}{x}\right)}

    the numerator approaches 1, while the denominator goes to infinity.

    That limit goes to zero, in which case the other also goes to zero or a negative value.
    Maybe you could examine that ?


    Hint: You could compare the limits of xlog_ex+1 and xe^x+1
    Last edited by Archie Meade; February 22nd 2011 at 06:19 AM. Reason: added a hint
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