Need help with proving conversion of a limit

[Jordanhuh]

Prove that lim t->0 (e^t-t-1)/t^2 = 1/2 without using L'Hospital Theorem.

tried everything:neutral::neutral::neutral::neutral:

 

[stapel]

Prove that lim t->0 (e^t-t-1)/t^2 = 1/2 without using L'Hospital Theorem.

I'm taking the limit to be as follows:

. . . . .\(\displaystyle \displaystyle \lim_{t\, \rightarrow\, 0}\, \dfrac{e^t\, -\, t\, -\, 1}{t^2}\, =\, \dfrac{1}{2}\)

(Sometimes formatting can be weird. If I've erred, kindly please reply with corrections.)

Certainly we can use l'Hospital's to confirm that the limit value is indeed 1/2. However, since you've posted this to "Algebra" rather than to "Calculus", I'm guessing that you're not even wanting to try log differentiation or anything; you're wanting to do this strictly algebraically.

tried everything:neutral::neutral::neutral::neutral:

Great! Please reply showing at least two of your different efforts. Thank you!