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Problem 27 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.

$ \displaystyle \lim_{x\to 0} \frac{e^x - 1 - x}{x^2} $

Answer

This limit has the form $\frac{0}{0} \cdot \lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}} \stackrel{\text { ? }}{=} \lim _{x \rightarrow 0} \frac{e^{x}-1}{2 x} \triangleq \lim _{x \rightarrow 0} \frac{e^{x}}{2}=\frac{1}{2}$

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Video Transcript

Okay. Find the limit you stuck up. Thus ruler a proper it. If there's a moral dimension, my third, consider using it. If locked US rule doesn't apply. Explain why. Okay, Here we need to find the limit off dysfunction. And first we can plug in zero to dysfunction that is e to the zero, which is one minus one and then minus zero. So the top zero and divided by plucky zero to hear that hiss. Dear square, which is general. So this is zero divided by zero. I am determined form so we can use the locked house rules to read it as the limit as X goes to zero that curative off the top, which is e to the X minus the curative off a number one that hiss Cyril. So minus zero and then minus the curative off acts that hiss one and then divided by the d word half off. Ask x square. That is too X. Okay, here we get a new function and you can Stu plugging zero that is e to the zero, which is one and minus one. He wanted by two times zero. That is zero. So you get another Neil Function zero divided by zero, which is also and determine the phone so we can use the lock house rule again and that his limit as axe goes to zero the curative off the top. That's just e to the X, then divided by the curative off the bottom, which is too so here for this functioning, plucking zero that hiss eat with zero it's one and divided by two, so the answer is a half.