Question

Prove by definition lim(n^2/(n^2+1))=1

Answer 1

n----> infinity ?

Answer 2

No just prove by definition

Answer 3

what kind of limit doesn't have bounds?

Answer 4

Just says prove directly by definition

Answer 5

so I have to apply that four step method., Increment
subtraction
Division
Limit..........

Answer 6

pretty sure this was done last night
let me see if i can find it

Answer 7

ok idea is this
you want to show that given any \(\epsilon>0\) you can find an N such that if \(n>N\) you have
\[|\frac{n^2}{n^2+1}-1|<\epsilon\]

Answer 8

algebra inside the absolute value signs gives
\[\frac{-1}{n^2+1}\] whose absolute value is \(\frac{1}{n^2+1}\) so you need to pick N with
\[\frac{1}{n^2+1}<\epsilon\] or
\[n^2+1>\frac{1}{\epsilon}\] or
\[n>\sqrt{\frac{1}{\epsilon-1}}\]

Answer 9

Okay thanks