Question

Find the smallest N such that when n>N, the absolute value of that big thingy < 1/10. Find the smallest "N" such that |[(6n)/(3n+7)]-2|< (1/10)...so basically its the absolute value of (6n)/(3n+7) then that whole quantity minus 2 is less than (1/10)... i think this is an epsilon type question but really need a push in the right direction! Thanks!!

Answer 1

I think what you want to do is find the value of the argument inside the absolute value term to be as small as possible. In order to do that, you want to solve the absolute value function so that it is zero basically. So |[(6n)/(3n+7)]-2|=0. Here you can just take away the absolute value signs and move the 2 over to the other side of the equation and you have [(6n)/(3n+7)]=2

Answer 2

Oh hold on, I think I interpreted your question the wrong way

Answer 3

does it help if i mention this? Find the smallest N such that when n>N, the absolute value of that big thingy < 1/10

Answer 4

\[\left| \frac{ 6n }{ 3n+7 }-2 \right|<\frac{ 1 }{ 10 }\]
\[\left| 2-\frac{ 14 }{ 3n+7 }-2 \right|<\frac{ 1 }{ 10 }\]
\[\left| -\frac{ 14 }{ 3n+7 } \right|<\frac{ 1 }{ 10 }\]
\[-\frac{ 1 }{ 10 }<-\frac{ 14 }{ 3n+7 } <\frac{ 1 }{ 10 }\]
\[\frac{ 1 }{ 140 }>\frac{ 1 }{ 3n+7 } >-\frac{ 1 }{ 140 }\]
\[140>3n+7>-140\]
\[133>3n>-147\]
\[\frac{ 133 }{ 3 }>n>-49\]
\[-49<n<\frac{ 133 }{ 3 }\]Now can u answer your question?

Answer 5

I took really good notes in class but when I got home, I didn't understand this...this closest thing I had was an epsilon delta proof...this is in the sequences section of my calc2 book i believe but I can't find an example of it! :o(

Answer 6

wow zecharias! that looks like you put alot of effort into helping me! thank you! let me look at it for a moment and see if i understand! :o)

Answer 7

i see that in step 2 it looks like you added 2 to the left side, but nothing changed on the right...i also am confused as to how 6n turned into 14? sorry i'm confused :o(

Answer 8

\[\frac{ 6n }{ 3n+7 }=2-\frac{ 14 }{ 3n+7 }\]

Answer 9

why do those two equations equal each other? i mean, what was the first thing that you thought of doing to even get to that step?

Answer 10

yeas i think so...i recently did some integration that i had to decompose stuff into smaller stuff so that i could integrate them...is that what you mean?

Answer 11

yes and can u apply that on\[\frac{ 6n }{ 3n+7 }\]

Answer 12

well i would need a minute to try...brb

Answer 13

hey Zek...im going to draw something...can you tell me if i'm on the right track? :o)

Answer 14

Don't do that. Just reach the step 2.