limit of sequence

missyc8

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Sep 7, 2009
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lim n ->infinity ((n+6)/(n+1) )^n

= (1+ 6/n) / 1 + 1/n)^n n->infinity .. ineterm. form

so, apply LH rule.. lim n-> infinity lny =n ln (n+6)/(n+1) = lim n -> infinity ln ((n+6)/(n+1) )/ (n^-1) = 0/ infinity ..indeterm form...

i get stuck here...you have to use chain rule? if yes..i am not getting the right answer
can someone help?
 
You do not need L'Hopital. Long divide your problem and get limn(1+5n+1)n\displaystyle \lim_{n\to \infty}\left(1+\frac{5}{n+1}\right)^{n}

Note that this is a famous 'e' limit. Note the silialrity with e=limn(1+1n+1)n\displaystyle e=\lim_{n\to \infty}\left(1+\frac{1}{n+1}\right)^{n}.

See your limit now?.
 
Hello, missyc8!

Galactus is absolutely correct,
. . and some Olympic-level gymnastics is required . . .


limn(n+6n+1)n\displaystyle \lim_{n\to\infty} \left(\frac{n+6}{n+1}\right)^n

We will use the definition:   limx(1+1x)x  =  e\displaystyle \text{We will use the definition: }\;\lim_{x\to\infty}\left(1 + \frac{1}{x}\right)^x \;=\;e


Using long division:   n+6n+1  =  1+5n+1\displaystyle \text{Using long division: }\;\frac{n+6}{n+1} \;=\;1 + \frac{5}{n+1}

. . Then:   (n+6n+1)n  =  (1+5n+1)n\displaystyle \text{Then: }\;\left(\frac{n+6}{n+1}\right)^n \;=\;\left(1 + \frac{5}{n+1}\right)^n


Multiply top and bottom by (1+5n+1) ⁣:    (1+5n+1)n11+5n+11+5n+1  =  (1+5n+1)n+11+5n+1\displaystyle \text{Multiply top and bottom by }\left(1 + \tfrac{5}{n+1}\right)\!:\;\;\frac{\left(1 + \frac{5}{n+1}\right)^n}{1}\cdot\frac{1 + \frac{5}{n+1}}{1 + \frac{5}{n+1}} \;=\;\frac{\left(1 + \frac{5}{n+1}\right)^{n+1}}{1 + \frac{5}{n+1}}

. . The numerator is:   (1+5n+1)n+1  =    [(1+5n+1)n+1]55  =    [(1+5n+1)n+15]5  =    [(1+1n+15)n+15]5\displaystyle \text{The numerator is: }\;\left(1 + \frac{5}{n+1}\right)^{n+1} \;=\;\;\left[\left(1 + \frac{5}{n+1}\right)^{n+1}\right]^{\frac{5}{5}} \;=\;\;\left[\left(1 + \frac{5}{n+1}\right)^{\frac{n+1}{5}}\right]^5 \;=\;\;\left[\left(1 + \frac{1}{\frac{n+1}{5}}\right)^{\frac{n+1}{5}}\right]^5


And we have:   limn[(1+1n+15)n+15]51+5n+1    =    lim[(1+1n+15)n+15]5lim(1+5n+1)\displaystyle \text{And we have: }\;\lim_{n\to\infty}\frac{\left[\left(1 + \frac{1}{\frac{n+1}{5}}\right)^{\frac{n+1}{5}}\right]^5}{1 + \frac{5}{n+1}} \;\;=\;\;\frac{\lim\left[\left(1 + \frac{1}{\frac{n+1}{5}}\right)^{\frac{n+1}{5}}\right]^5} {\lim\left(1 + \frac{5}{n+1}\right)}

. . . . . . . . =    [lim(1+1n+15)n+15]5lim(1+5n+1)    =    e51+0    =    e5\displaystyle =\;\;\frac{\left[\lim\left(1 + \frac{1}{\frac{n+1}{5}}\right)^{\frac{n+1}{5}}\right]^5} {\lim\left(1 + \frac{5}{n+1}\right)} \;\;=\;\;\frac{e^5}{1+0} \;\;=\;\;e^5

. . Whew! . . . I need a nap.

 
However you could use the well known corollary - and avoid hurtful gymnastics:

limn(1+xn)n  =  ex\displaystyle \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^n \;=\;e^x
 
Another way.

Find limn(n+6n+1)n, Let y = limn(n+6n+1)n\displaystyle Find \ \lim_{n\to\infty}\bigg(\frac{n+6}{n+1}\bigg)^{n}, \ Let \ y \ = \ \lim_{n\to\infty}\bigg(\frac{n+6}{n+1}\bigg)^{n}

Then, lny = ln[limn(n+6n+1)n].\displaystyle Then, \ ln|y| \ = \ ln\bigg[\lim_{n\to\infty}\bigg(\frac{n+6}{n+1}\bigg)^{n}\bigg].

lny = limnln(n+6n+1)n = limnnln(n+6n+1)\displaystyle ln|y| \ = \ \lim_{n\to\infty}ln\bigg(\frac{n+6}{n+1}\bigg)^{n} \ = \ \lim_{n\to\infty}nln\bigg(\frac{n+6}{n+1}\bigg)

limnnln(n+6n+1) gives indeterminate form   0, however, not to despair.\displaystyle \lim_{n\to\infty}nln\bigg(\frac{n+6}{n+1}\bigg) \ gives \ indeterminate \ form \ \infty \ * \ 0, \ however, \ not \ to \ despair.

limnln(n+6n+1)1n gives indeterminate form 00.\displaystyle \lim_{n\to\infty}\frac{ln\bigg(\frac{n+6}{n+1}\bigg)}{\frac{1}{n}} \ gives \ indeterminate \ form \ \frac{0}{0}.

Therefore, the Marqui to the rescue gives, limnnln(n+6n+1) = 5.\displaystyle Therefore, \ the \ Marqui \ to \ the \ rescue \ gives, \ \lim_{n\to\infty}nln\bigg(\frac{n+6}{n+1}\bigg) \ = \ 5.

Hence, lny = 5,      y = e5.\displaystyle Hence, \ ln|y| \ = \ 5, \ \implies \ y \ = \ e^{5}.

Therefore, after all is said and done, limn(n+6n+1)n = e5.  QED\displaystyle Therefore, \ after \ all \ is \ said \ and \ done, \ \lim_{n\to\infty}\bigg(\frac{n+6}{n+1}\bigg)^{n} \ = \ e^{5}.\ \ QED
 
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