Undefined Limit - e^x-1/x^2 as x Approaches 0?

[dan38]

Homework Statement

Don't understand why the limit (e^x - 1)/x^2 is undefined as x approaches 0?
Can't you use L'hopital's rule to get the value as 1/2?

Homework Equations

The Attempt at a Solution

 

[micromass]

l'hospital's rule certainly applies here. But I don't see how you use that to get 1/2. Perhaps you can provide your work?

 

[dan38]

(e^x - 1) / x^2
Differentiate top and bottom
e^x / 2x
Differentiate top and bottom
e^x / 2
Sub in x = 0
1/2?

 

[algebrat]

you applied l'Ho(s)pital's rule the second time, when it was not of the indeterminate form. What is the limit of e^x/(2x)?

 

[Muphrid]

I'm not sure why, but I suspect there may be a subtlety here in how l'Hopital's rule can be applied because l'Hopital should be a simple shortcut in lieu of a Taylor series, yet a Taylor series analysis yields (e^x-1)/x^2 -> (x + x^2/2 + ...)/x^2 -> 1/x + 1/2.

 

[dan38]

I don't understand algebrat, you can't divide by zero?

 

[Muphrid]

He's saying that l'Hopital's rule only applies when you have an indeterminate form: things like $0 \times \infty$ are indeterminate, as is $0/0$, but in the context of limits, $1/0$, while undefined, is not indeterminate. $1/0$ either gives $\pm \infty$ or the limit does not exist, as in this case.

 

[micromass]

Indeed, l'hospital's rule only applies to situations like

$$\frac{0}{0}~\text{and}~\frac{\pm \infty}{\pm \infty}$$

Now the situation is $\frac{1}{0}$, so we can not appy l'hospital's rule.

 

[dan38]

oh i see, thanks!
doing a question: x^2*e^x with x---> infinity
changed it to
x^2/e^(-x)
used L'hopital's rule
2x/-e^(-x)
and again
2/e^(-x)
where do I go from here? :S

 

[Muphrid]

You've missed the point. You can't just apply l'Hopital's rule whenever you want. You can only apply it when the functions in the limit would evaluate to something indeterminate. After you applied l'Hopital once, you got, essentially, $\infty/0$. This is not indeterminate. You can't apply l'Hopital again.

Honestly, I would recommend abandoning l'Hopital's rule and just Taylor expanding things. It may require a little extra crunching, but it's a lot more unambiguous and idiot-proof.

 

[Ratch]

dan38,

Don't understand why the limit (e^x - 1)/x^2 is undefined as x approaches 0?
Can't you use L'hopital's rule to get the value as 1/2?

The limit of the term as x-->0 is not defined because it is +∞ when x-->0+ and -∞ when x-->0- . The limit has to be defined in order to use L'Hospital's rule.

Ratch

 

[dan38]

2x/-e^(-x)
so how would I go from here then?

 

[clamtrox]

Plug in the value x=0.

 

[HallsofIvy]

Indeed, l'hospital's rule only applies to situations like

$$\frac{0}{0}~\text{and}~\frac{\pm \infty}{\pm \infty}$$

oh i see, thanks!
doing a question: x^2*e^x with x---> infinity
changed it to
x^2/e^(-x)

As x goes to infinity, the numerator goes to infinity and e^-x goes to 0. micromass had just told you that you cannot use l'hospital's rule for that. And you don't need to. The numerator of a fraction going to infinity while the denominator goes to 0, the fraction goes to infinity. Actually, that should have been obvious from the start. $x^2e^x$ is the product of two functions both of which go to infinity.

Using L'hopital's rule
2x/-e^(-x)
and again
2/e^(-x)
where do I go from here? :S

 

[Ratch]

dan38,

2x/-e^(-x)
so how would I go from here then?

You don't go anywhere. That function's limit is undefined. It has a one value when x-> is approached from the left, and another value when approached from the right. Isn't that what I said earlier?

Ratch