Solving a Limit: Evaluating limx->0 (e^x - 1- x - (x^2/2))/x^3

[lab-rat]

Homework Statement

Evaluate limx->0 (e^x - 1- x - (x^2/2))/x^3

The Attempt at a Solution

I can't remember how to solve this limit. Do I need to evaluate each part seperately? I plugged in the 0 to find that the limit does exist. I just can't seem to figure out what to do next.

 

[BloodyFrozen]

Why not use l'Hôpital's rule?

 

[micromass]

Or use the Taylor series of the exponential function.

 

[lab-rat]

If I use l'Hopital's rule won't I end up with 3x^2 at the bottom?

 

[micromass]

You will want to use l'Hopital multiple times.

 

[genericusrnme]

The limit of a sum is the sum of limits - Lim[a+b]=Lim[a]+Lim
Then as everyone before has said, you're going to want to use l'hopitals rule for the last term

 

[lab-rat]

I used L'Hopital's rule three times and ended up with limx->0(e^x)/5

Is this correct? It still gives me 0 on top.

 

[lab-rat]

Here is how I did it in case it might help

lim->0 (e^x)' - 1' - x' - (x^2/2)' /x^3'

= lim->0 (e^x' - 1' - x')/ 3x^2'

= lim->0 (e^x' - 1') / 5x'

= lim x->0 (e^x) /5

 

[BloodyFrozen]

Very close, but look at your second to thrid line in the denominator.

What is: $$\frac{d}{dx}(3x^{2})$$

After you fix that, evaluate it at 0.

$$\lim_{x\to 0}~ e^{x}$$ should not be 0.

 

[lab-rat]

Thanks!
It should be 6x right?

So is 1/6 the correct answer?

 

[SammyS]

1/6 is the correct result.