Help with limit evaluation

Asked Dec 08 '17 at 06:44

Active Dec 08 '17 at 06:44

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Okay so I have to find this limit

$$\lim_{x\rightarrow 0}\frac{(1+x)^\frac{1}{x}-e+\frac{ex}{2}}{11x^2}$$

I could have used the expansion of $(1+x)^\frac{1}{n}$ had $n$ been an integral value , but since it approaches 0 in this case , what are my best options to solve this limit ?

$(1+x)^\frac{1}{x}$ turns to an indeterminate form of $1^{\infty}$.Can I separately solve it and put its limiting value in my question above ?

asked Dec 08 '17 at 06:44

Tanuj

I asked a very similar question just few minutes ago. Look at the taylor series of your function. – jonsno Dec 08 '17 at 06:47
https://math.stackexchange.com/questions/2556766/limit-as-x-to-0-of-frac1x1-x-ex – jonsno Dec 08 '17 at 06:47
Wow thanks ! I feel stupid now ! – Tanuj Dec 08 '17 at 06:48
Yeah me too feel stupid. – jonsno Dec 08 '17 at 06:49
Are you by any chance preparing for the JEE ? – Tanuj Dec 08 '17 at 06:50
Yeah, but aren't most stupids doing that :) – jonsno Dec 08 '17 at 06:52
Yeah , me too bro ! What books do you solve ? – Tanuj Dec 08 '17 at 06:53
I recommend you to come on this chat: PSS – jonsno Dec 08 '17 at 06:55
No one here has to feel stupid ! – Claude Leibovici Dec 08 '17 at 06:57
This is best handled by Taylor series. You can just try to get the Taylor series via $(1+x)^{1/x}=\exp(1-x/2+x^2/3+\dots)$ which leads to $e(1-x/2+11x^2/24+\dots)$. – Paramanand Singh Dec 08 '17 at 15:08
Thank you sir , I figured it out . I wish I could ask you all of my doubts. Can this be made possible anyhow ? – Tanuj Dec 08 '17 at 15:20

Help with limit evaluation

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