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I've got problems with calculating the limits in these two examples:

$$\begin{align*} &\lim_{ n \to +\infty}\left( \sqrt{n}\cdot \left( e^{\frac{1}{\sqrt{n}}} -2^{\frac{1}{\sqrt{n}}} \right) \right)^3\\ &\lim_{n \to +\infty} n\cdot \sqrt{e^{\frac1n}-e^{\frac{1}{n+1}}} \end{align*}$$

Can anybody help?

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What kind of problems? Where do you get stuck? –  Andy Mar 8 '12 at 14:28
    
Both are consequences of the fact that $\mathrm e^x-1=x+o(x)$ when $x\to0$, the first one for $x=(1-\log2)/\sqrt{n}$ and the second one for $x=1/(n(n+1))$. –  Did Mar 8 '12 at 14:29
    
@Andy, I don't even know how to start and the tip above doesn't say anything to me. –  xan Mar 8 '12 at 14:32

5 Answers 5

up vote 2 down vote accepted

For the first one, let $u=\frac1{\sqrt n}$, and note that $u\to 0^+$ as $n\to\infty$. Then $$\sqrt n\left(e^{\frac1{\sqrt n}}-2^{\frac1{\sqrt n}}\right)=\frac1u\left(e^u-2^u\right)=\frac{e^u-2^u}u\;,$$ so you’re interested in $$\lim_{u\to 0^+}\frac{(e^u-2^u)^3}{u^3}\;,$$ and I expect that you know a way to deal with that kind of limit.

I can make a similar trick work for the second one, but it gets a bit messier. First, I’m actually going to look at $$\lim_{n\to\infty}n^2\left(e^{\frac1n}-e^{\frac1{n+1}}\right)$$ and then take its square root to get the desired limit.

Let $u=\frac1n$, so that $n=\frac1u$, $n+1=\frac1u+1=\frac{u+1}u$, and $\frac1{n+1}=\frac{u}{u+1}=1-\frac1{u+1}$. Again $u\to 0^+$ as $n\to\infty$, so I look at $$\lim_{u\to 0^+}\frac{e^u-e^{1-\frac1{u+1}}}{u^2}\;;$$ applying l’Hospital’s rule takes a little more work this time, but it’s still eminently feasible.

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L'Hôpital's rule? –  xan Mar 8 '12 at 14:52
    
@xan: That’s right. –  Brian M. Scott Mar 8 '12 at 14:58
    
ok :-) Does the second one can be solved with similar approach? I can't see there substitution like in first one. –  xan Mar 8 '12 at 15:02
    
@xan: It can indeed, and I’ve added to my answer to get you started. –  Brian M. Scott Mar 8 '12 at 15:05
    
great! thank you very much :-) –  xan Mar 8 '12 at 15:07

What Didier is suggesting the use of the little oh notation. This stands for the following:

$$e^x=1+x+o(x) \text{ for } x\to 0 $$ means that

$$\lim\limits_{x \to 0} \frac{e^x-1-x}{x} =0$$

or that we can approximate $e^x$ near $x=0$ very accurately by $1+x$, with an error that is very small in comparison to $x$ i.e $\dfrac{o(x)}{x}\to0$ (that is what $o(x)$ kinda means.)

To make things easier, put $\sqrt n=x$, to get

$$\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{e^x} - {2^x}}}{x}} \right)^3}$$

Now, we get the following:

$$\eqalign{ & {e^x} = 1 + x + o\left( x \right) \cr & {2^x} = {e^{x\log 2}} = 1 + x\log 2 + o\left( x \right) \cr} $$

So this means:

$$\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{1 + x + o\left( x \right) - 1 - x\log 2 - o\left( x \right)}}{x}} \right)^3}$$

$$\mathop {\lim }\limits_{x \to 0} {\left( {1 - \log 2 + \frac{{o\left( x \right)}}{x}} \right)^3}$$

But since we said $o(x)$ is an error that tends to zero in comparison to $x$, we get

$$\mathop {\lim }\limits_{x \to 0} {\left( {1 - \log 2 + \frac{{o\left( x \right)}}{x}} \right)^3} = {\left( {1 - \log 2} \right)^3}$$

Do you follow? Can you move on to the second case analogously?

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This is more or less Brian M. Scott's answer, without converting to a limit at $0^+$.

For the second limit, let's find the limit of the square. Using L'Hôpital's Rule twice: $$\eqalign{ \lim_{n\rightarrow\infty} n^2(e^{1\over n}-e^{1\over n+1} ) &=\lim_{n\rightarrow\infty} {e^{1\over n}-e^{1\over n+1} \over 1/n^2}\cr &=\lim_{n\rightarrow\infty} {(-1/n^2)e^{1/n}-(-1/(n+1))^2e^{1\over n+1} \over-2/n^3}\cr &=\lim_{n\rightarrow\infty} { e^{1\over n}-({n\over n+1})^2e^{1\over n+1} \over2 /n}\cr &=\lim_{n\rightarrow\infty} {(-1/n^2) e^{1\over n}-\Bigl[\,({n\over n+1})^2 \cdot{-1\over (n+1)^2}e^{1\over n+1}+{2n\over (n+1)^3}e^{1\over n+1}\Bigr] \over-2 /n^2}\cr &=\lim_{n\rightarrow\infty} { e^{1\over n}-({n\over n+1})^4 e^{1\over n+1} + {2n^3\over (n+1)^3}e^{1\over n+1} \over2 }\cr &={1-1+2\over 2}\cr &=1. } $$ So, the limit of the original expression is $\sqrt 1=1$.


For the first limit, I would initially ignore the cube, and first compute $$ \lim_{n\rightarrow\infty} \bigl[\,{\sqrt n(e^{1/\sqrt n}- 2^{1/\sqrt n} )}\,\bigr] =\lim_{n\rightarrow\infty}{ { e^{1/\sqrt n}- e^{\ln 2/\sqrt n} }\over 1/\sqrt n }. $$ One application of L'Hôpital's Rule will show the above limit is $1-\ln 2$; whence the original limit is $(1-\ln 2)^3$.

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Because the limit of the cube of a sequence is the cube of the limit, it suffices to find and cube the following limit $$\lim_{n \rightarrow \infty} \sqrt{n}(e^{1 \over \sqrt{n}} - 2^{1 \over \sqrt{n}})$$ Note that since $n$ going to infinity is the same as ${1 \over \sqrt{n}}$ going to zero, the above limit is $$\lim_{x \rightarrow 0} {e^{x} - 2^{x} \over x}$$ L'hopital's rule gives that this limit is $1 - \ln2$, so the original limit is $(1 - \ln 2)^3$.

You can do something similar with the second one, but it's more complicated. The limit is the square root of $$\lim_{n \rightarrow \infty} n^2(e^{1 \over n} - e^{1 \over n + 1})$$ Letting $x = {1 \over n}$ this becomes $$\lim_{x \rightarrow 0} {e^x - e^{x \over x + 1} \over x^2}$$ Since $\lim_{x \rightarrow 0} e^x = 1$, you can factor out an $e^x$ in the above to obtain $$\lim_{x \rightarrow 0} {1 - e^{-{x^2 \over x + 1}} \over x^2}$$ Since $x +1$ goes to $1$, this is the same as $$\lim_{x \rightarrow 0} {1 - e^{-{x^2 \over x + 1}} \over {x^2 \over x+1}}$$ So if you let $y = {x^2 \over x + 1}$ the above becomes $$\lim_{y \rightarrow 0} {1 - e^{-y} \over y}$$ By L'hopital's rule (or other methods) this limit is $1$, so the original limit is the square root of this, also $1$.

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This answer is useful. –  The Chaz 2.0 Mar 9 '12 at 4:15

Let's solve them elementarily:

1st limit: $$\lim_{ n \to +\infty}\left( \sqrt{n}\cdot \left( e^{\frac{1}{\sqrt{n}}} -2^{\frac{1}{\sqrt{n}}} \right) \right)^3=\lim_{ n \to +\infty}\left(\frac{e^{\frac{1}{\sqrt{n}}}-1}{\frac{1}{\sqrt{n}}} - \frac{2^{\frac{1}{\sqrt{n}}}-1}{\frac{1}{\sqrt{n}}}\right)^3=(1-\ln2)^{3}.$$ 2nd limit: $$L^2=\lim_{n \to +\infty} \left(n\cdot \sqrt{e^{\frac1n}-e^{\frac{1}{n+1}}}\right)^2=\lim_{n \to +\infty}n^2\left(\frac{e^{\frac{1}{n}}-1}{\frac{1}{n}}\cdot\frac{1}{n} - \frac{e^{\frac{1}{n+1}}-1}{\frac{1}{n+1}}\cdot\frac{1}{n+1}\right)$$ $$=\lim_{n \to +\infty} \frac{n^2}{n(n+1)}=1.$$ Therefore, $L=1$.

Remark: for the two limits i resorted to the auxiliary limit, $\lim_{x\to0} \frac{a^x-1}{x}=\log a$, $a>0$.

Q.E.D.

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You can say that $$\log x = \lim\limits_{k \to 0}\frac{x^k-1}{k}$$ so that it is clearer. –  Pedro Tamaroff Jun 19 '12 at 16:17
    
One little other thing: QED means "which was to be demonstrated", so you can start by saying "The limit is $[\text{blah}]$" and then prove it so the QED makes sense. –  Pedro Tamaroff Jun 19 '12 at 16:19
    
I don't understand what you mean. –  Chris's sis Jun 19 '12 at 16:35
    
You end by saying "which was to be proven" but don't start with, "The limit is $X$". Just a matter of structure. –  Pedro Tamaroff Jun 19 '12 at 16:37
    
@Peter Tamaroff: oh, it's an automatism i learned from my teacher. –  Chris's sis Jun 19 '12 at 16:38

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