Question

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)?

Answer 1

The general solution amounts to applying l'Hopital's rule. Are you familiar?

Answer 2

No Im not. Can you show me?

Answer 3

l'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,\[\lim_{x\to0}\frac{\sin x}{x}=\lim_{x\to0}\frac{\frac{d}{dx}\sin(x)}{\frac{d}{dx}x}=\lim_{x\to0}\frac{\cos x}{1}=1\]

Answer 4

Oh. So essentially you're calculating the derivative of the initial problem and then finding the limit of that?

Answer 5

You're finding the derivative of numerator and denominator (each evaluated separately), application-wise. The limits should be the same. This only applies when they normally evaluate to the indeterminate forms \(\frac00,\frac\infty\infty\).

Answer 6

So 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?

Answer 7

Okay, I'll be less technical; sorry, been reading a textbook recently. Here's a practice problem for you.\[\lim_{x\to0}\frac{e^x-1}{x}=\,\,?\]

Answer 8

Okay. 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^x-1 as one equation and the limit as it approaches 0 x, as the two problems?

Answer 9

No, I'll break it down for you.\[\lim_{x\to0}\frac{e^x-1}{x}\]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 \(\lim_{x\to0}e^x-1=\lim_{x\to0}x=0\text{ or }\infty\). Plugging in zero, as we just did, shows that this is the case. So, next step: apply the derivative of top and bottom.

Answer 10

Im sorry D: Im still a bit confused :/

Answer 11

Okay wait let me try continuing on paper. That explanation helped a bit.

Answer 12

\[\lim_{x\to0}\frac{e^x-1}{x}\to\lim_{x\to0}\frac{\frac{d}{dx}(e^x-1)}{\frac{d}{dx}x}\]

Answer 13

brb quick lunch. @Ishaan94 @eliassaab help here please

Answer 14

Amanderp? Is it still confusing you?

Answer 15

YES :(

Answer 16

What part is confusing you?

Answer 17

Im 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..

Answer 18

So I guess the rule itself, including the application is confusing me.

Answer 19

Hmm 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.

Answer 20

Im sorry. I need a refresh. Differentiation is simply taking the derivative right?

Answer 21

Yes.

Answer 22

Okay so then the answer to this problem that badreferences provided would be the derivative of the top and bottom, in fraction form?

Answer 23

Yes.

Answer 24

Like it wouldnt be evaluated, so to speak, with the 0. It would be left in terms of the variables?

Answer 25

It 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.

Answer 26

Okay 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.

Answer 27

I am sorry. I am not really a good teacher. I hope @eliassaab can help you.

Answer 28

I 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.

Answer 29

Lol I appreciate your honesty.
Im not a math person so its just not clicking :3

Answer 30

To 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}{ x-7}=\frac 0 0\\

lim_{x\to 7} \frac {x^2 - 8 x +7}{ x-7}=lim_{x\to 7} \frac {(x-7)(x-1)}{ x-7}=\lim_{x\to 7}(x-1)=6
\]

Answer 31

Ohmygod thats ten times easier. Can you do that with all problems that would typically require the rule?

Answer 32

no, 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

Answer 33

\[\sin x \approx x\]For lower values of x. Maybe that's why \[\lim_{x\to0} \frac{\sin x}x = 1\]

Answer 34

An approximation of \(\sin x\) at infinitesmal values around \(0\) reveals a slope of \(\cos0=1\therefore P(x)=x\).

Answer 35

Actually, 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

Answer 36

See also this
http://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions

Answer 37

Interesting. Thanks for the correction. :)