Problem

Find the limit. Use l'Hospital's Rule where appro…

09:00

Problem 15 Easy Difficulty

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.
$\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}}$

Answer

$\lim _{x \rightarrow \infty} \frac{e^{x}}{2}=\frac{e^{0}}{2}=\frac{1}{2}$

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Chapter 4

Applications of Derivatives

Section 3

L'Hospital's Rule: Comparing Rates of Growth

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Video Transcript

already. So here we have the eliminates X approaches zero of either the X minus one minus X over X squared. So we will start by evaluating the limit. And it's gonna be eat of the power of zero minus one minus zero over zero squared Eso. This, of course, is equal to one minus one minus 0/0, which is gonna be 0/0. So we can file a Beatles rule. And I know that this original limit will be equal to the limit as X approaches zero of the derivative of this top limit. So the derivative of E to the X is either the X, the derivative of minus 10 and the derivative of minus X is gonna be a minus one and then take that over the derivative of X squared and that will be two X. So now we're going to evaluate this limit, and in that case, we get E to the zero minus one over two times zero says could be one minus 1/0, and again, we have 0/0 so we can take low petals. Rule a second time. Okay, It's when we do that, we get that. This is gonna be equal to the limit. As X approaches zero the derivative of either the exes. Just eat of the ex. The derivative of the one is gonna be zero So far numerator We have just e the X and then when we take the derivative of two X, that's gonna be equal to two. So now we can evaluate the limit again, we get either the power of zero over to just gonna be equal to 1/2 and that is the solution or limit will be.

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