(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[itex]

\displaystyle\lim_{x\rightarrow 0^+} \frac{e^x - (1 + x)}{x^n}

[/itex]

where n is a positive integer

3. The attempt at a solution

The numerator approaches 0 as x approaches 0 from the right. The denominator approaches 0 with whichever positive integer, n. This gives an indeterminate form, so I can apply L'Hopital's Rule:

[itex]

\displaystyle\lim_{x\rightarrow 0^+} \frac{e^x - 1}{nx^{n-1}}

[/itex]

Now, here's where I'm not too sure...

The numerator will still approach 0. The denominator will still approach 0 as x approaches 0, it's just being multiplied by some constant. If I apply L'Hopital's Rule again:

[itex]

\displaystyle\lim_{x\rightarrow 0^+} \frac{e^x}{n(n-1)x^{n-2}}

[/itex]

The numerator now reaches 1 as x approaches 0.

I'm unsure of how to evaluate the limit because I'm unsure of how to treat the denominator.

If n was some large positive integer, I'd just get (1/0) as x approaches 0 which would yield to infinity... But if n had been some very small integer, the denominator would have eventually simplified to some constant term and evaluating the limit would not give infinity.

How do I actually go on to evaluate the limit without assigning a value for n?

Would it be legal to just specify the two specific cases (n = 1 and n = 2) where the limit doesn't evaluate towards infinity and then just give the general case for n > 2?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Limit involving L'Hopital's Rule

**Physics Forums | Science Articles, Homework Help, Discussion**