Question

use the formal definition of limit to verify the indicated limit:

limit x-->infinite (1/sqrt(x))

Answer 1

Pick \[N=\frac{1}{\epsilon^2}\]

Answer 2

oops, limit x-->infinite (1/sqrt(x)) = 0

Answer 3

Obviously ;)

Answer 4

let \[\epsilon\] be a given positive number. for x > 0 we have
| 1/sqrt(x) - 0 | = 1/|sqrt(x)|, 1/x < \[\epsilon^2\] provided x>1/\[\epsilon^2\]
R = 1/e^2 => R^(1/2)=1/e

???

Answer 5

is it wrong? :(?

Answer 6

yes..

\[\text{Let }\epsilon>0\]
\[\text{Choose }N=\frac{1}{\epsilon^2}\]

then for all \[x>N\] we have

\[x>\frac{1}{\epsilon^2}\]

\[\Rightarrow \epsilon^2>\frac{1}{x}\]

\[\Rightarrow \epsilon>\frac{1}{\sqrt{x}}\]