How to find specified limits

Rays

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\(\displaystyle \displaystyle \mbox{1. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{2n^3\, +\, n\, -\, 3}{(2n\, -\, 1)^3}\)

\(\displaystyle \displaystyle \mbox{2. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{(n\, +\, 1)\,(n\, +\, 2)\,(n\, +\, 3)}{n^3\, +\, 2n\, +\, 1}\)

\(\displaystyle \displaystyle \mbox{3. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{(3n\, -\, 1)\, (1\, -\, 2n)}{(2n\, +\, 3)\, (n\, -\, 3)}\)


Hey guys,

I know these examples are simple but i tried to solve them for about an hour and haven't succeed. Can someone help me?

Thank you in advance!
 
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Subhotosh Khan

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\(\displaystyle \displaystyle \mbox{1. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{2n^3\, +\, n\, -\, 3}{(2n\, -\, 1)^3}\)

\(\displaystyle \displaystyle \mbox{2. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{(n\, +\, 1)\,(n\, +\, 2)\,(n\, +\, 3)}{n^3\, +\, 2n\, +\, 1}\)

\(\displaystyle \displaystyle \mbox{3. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{(3n\, -\, 1)\, (1\, -\, 2n)}{(2n\, +\, 3)\, (n\, -\, 3)}\)


Hey guys,

I know these examples are simple but i tried to solve them for about an hour and haven't succeed. Can someone help me?

Thank you in advance!
\(\displaystyle \displaystyle \mbox{1. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{2n^3\, +\, n\, -\, 3}{(2n\, -\, 1)^3}\)

For the first problem divide the numerator and the denominator by n3 - then apply the limit.
 
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stapel

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i tried to solve them for about an hour and haven't succeed.
Great! Then you've got loads of thoughts and efforts that you can display, so we can try to figure out where things are going sideways.

\(\displaystyle \displaystyle \mbox{1. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{2n^3\, +\, n\, -\, 3}{(2n\, -\, 1)^3}\)
You noted what the leading term of the denominator would have to be, after expansion. You applied the formula they gave you for finding horizontal asymptotes (that is, the "value" "as x tends toward infinity"). (refresher) And... then what?

\(\displaystyle \displaystyle \mbox{2. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{(n\, +\, 1)\,(n\, +\, 2)\,(n\, +\, 3)}{n^3\, +\, 2n\, +\, 1}\)
This one works in exactly the same way. What did you get, after you simplified the formula's ratio of leading terms?

\(\displaystyle \displaystyle \mbox{3. }\, \lim_{n\, \rightarrow\, \infty}\, \dfrac{(3n\, -\, 1)\, (1\, -\, 2n)}{(2n\, +\, 3)\, (n\, -\, 3)}\)
You multiplied out the two products, you divided the leading terms, and... then what?

Please be complete, showing all of the methods you attempted to apply. Thank you! ;)
 

Rays

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Sep 19, 2015
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Thank you guys! With your help i solved these :)
 
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