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Amanderp
 2 years ago
How does one handle the case of a limit that is in indeterminate form (turns out to be 0/0 when the number is plugged in)?
Amanderp
 2 years ago
How does one handle the case of a limit that is in indeterminate form (turns out to be 0/0 when the number is plugged in)?

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badreferences
 2 years ago
Best ResponseYou've already chosen the best response.5The general solution amounts to applying l'Hopital's rule. Are you familiar?

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0No Im not. Can you show me?

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.5l'Hopital's rule asserts that, for any \(\lim_{x\to a}f(x)=\lim_{x\to a} g(x)=0,\infty\), the equivalence\[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}\]can be established. For instance,\[\lim_{x\to0}\frac{\sin x}{x}=\lim_{x\to0}\frac{\frac{d}{dx}\sin(x)}{\frac{d}{dx}x}=\lim_{x\to0}\frac{\cos x}{1}=1\]

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Oh. So essentially you're calculating the derivative of the initial problem and then finding the limit of that?

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.5You're finding the derivative of numerator and denominator (each evaluated separately), applicationwise. The limits should be the same. This only applies when they normally evaluate to the indeterminate forms \(\frac00,\frac\infty\infty\).

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0So then this is sort of like the quotient rule in the sense that you have to break up the function into two separate problems and then evaluate to get the answer?

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.5Okay, I'll be less technical; sorry, been reading a textbook recently. Here's a practice problem for you.\[\lim_{x\to0}\frac{e^x1}{x}=\,\,?\]

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Okay. So. This is going to come out to 0/0 so you have to break it up right? So itll be the limit as it approaches 0 e^x1 as one equation and the limit as it approaches 0 x, as the two problems?

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.5No, I'll break it down for you.\[\lim_{x\to0}\frac{e^x1}{x}\]is our problem. We note that this evaluates to the indeterminate form \(\frac00\) by just plugging in \(0\) for \(x\). So, let's instead apply l'Hopital's rule. First, we need to make sure we meet the prerequisites: that \(\lim_{x\to0}e^x1=\lim_{x\to0}x=0\text{ or }\infty\). Plugging in zero, as we just did, shows that this is the case. So, next step: apply the derivative of top and bottom.

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Im sorry D: Im still a bit confused :/

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Okay wait let me try continuing on paper. That explanation helped a bit.

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.5\[\lim_{x\to0}\frac{e^x1}{x}\to\lim_{x\to0}\frac{\frac{d}{dx}(e^x1)}{\frac{d}{dx}x}\]

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.5brb quick lunch. @Ishaan94 @eliassaab help here please

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.1Amanderp? Is it still confusing you?

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.1What part is confusing you?

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Im not sure. Like I know how to find the limit of something. But my teacher never actually went over this rule. And Im not really understanding it..

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0So I guess the rule itself, including the application is confusing me.

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.1Hmm what you are supposed to do is simply differentiate the numerator and denominator, if the limit is of zero by zero or infinity by infinity form.

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Im sorry. I need a refresh. Differentiation is simply taking the derivative right?

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Okay so then the answer to this problem that badreferences provided would be the derivative of the top and bottom, in fraction form?

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Like it wouldnt be evaluated, so to speak, with the 0. It would be left in terms of the variables?

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.1It would be evaluated at the given limit. 1. Check the form. 2. If it's zero by zero or infinity by infinity, you apply L'Hospital. 3. Evaluate it for the given limit.

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Okay but see thats what Im not sure about. Like I now kind of understand it, but I need a simpler definition of that rule. Because Im really not grasping it entirely.

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.1I am sorry. I am not really a good teacher. I hope @eliassaab can help you.

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.1I won't lie to you. Personally, I don't know why L'Hospital is used and what happens when you differentiate the numerator and denominator. For me it's more of a trick to get the limits quickly and I have always avoid using it.

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Lol I appreciate your honesty. Im not a math person so its just not clicking :3

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.0To avoid using L'Hospital's rule, you can sometimes factor up and down and cancel the term that is making up and down zero. \[ \lim_{x\to 7} \frac {x^2  8 x +7}{ x7}=\frac 0 0\\ lim_{x\to 7} \frac {x^2  8 x +7}{ x7}=lim_{x\to 7} \frac {(x7)(x1)}{ x7}=\lim_{x\to 7}(x1)=6 \]

Amanderp
 2 years ago
Best ResponseYou've already chosen the best response.0Ohmygod thats ten times easier. Can you do that with all problems that would typically require the rule?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0no, only in certain cases consider\[\lim_{x\to0}{\sin x\over x}\]this limit can be done with l'Hospital, but there is clearly no way to factor anything

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.1\[\sin x \approx x\]For lower values of x. Maybe that's why \[\lim_{x\to0} \frac{\sin x}x = 1\]

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.5An approximation of \(\sin x\) at infinitesmal values around \(0\) reveals a slope of \(\cos0=1\therefore P(x)=x\).

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.0Actually, the limit of sin(x)/x is used to prove that the derivative of sin(x) is cos(x). Technically, one should not use L'Hospital's rule to find the limit. See the proof on http://www.proofwiki.org/wiki/Limit_of_Sine_of_X_over_X/Geometric_Proof

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.0See also this http://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.5Interesting. Thanks for the correction. :)
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