- #1
RPierre
- 10
- 0
**DISCLAIMER - I am super bad at LaTeX**
Prove
[tex]\lim_{x \rightarrow \infty}\frac{1}{1+x^2} = 0[/tex]
I Think I proved it, but I feel like I'm missing something to make this a proof of ALL [tex]\epsilon[/tex]>0 and not just one case. Maybe I did it right. I really don't know. Just looking for a second opinion and/or advice on [tex]\epsilon[/tex]-[tex]\delta[/tex] proofs.
3.Attempt at a Solution
the definition logically is if [tex]x > N[/tex] then [tex]|f(x) - L| > \epsilon[/tex] for some [tex]N,\epsilon > 0[/tex]
Setting [tex]N=\sqrt{\frac{1-\epsilon}{\epsilon}}[/tex]
x > N
[tex]\Rightarrow[/tex] [tex] x > \sqrt{\frac{1-\epsilon}{\epsilon}}[/tex]
[tex]\Rightarrow[/tex] [tex] x^2 > \frac{1-\epsilon}{\epsilon}[/tex]
[tex]\Rightarrow[/tex] [tex] x^2 + 1 > \frac{1}{\epsilon}[/tex]
[tex]\Rightarrow[/tex] [tex] \frac{1}{1+x^2} < \epsilon [/tex]
[tex] \frac{1}{1+x^2} = f(x) [/tex]
and since N > 0, and x > N, it is implied x > 0 and therefore [tex]|f(x)| = f(x)[/tex]
I'm not sure if this is a good enough proof? Thanks in Advance :)
Homework Statement
Prove
[tex]\lim_{x \rightarrow \infty}\frac{1}{1+x^2} = 0[/tex]
Homework Equations
I Think I proved it, but I feel like I'm missing something to make this a proof of ALL [tex]\epsilon[/tex]>0 and not just one case. Maybe I did it right. I really don't know. Just looking for a second opinion and/or advice on [tex]\epsilon[/tex]-[tex]\delta[/tex] proofs.
3.Attempt at a Solution
the definition logically is if [tex]x > N[/tex] then [tex]|f(x) - L| > \epsilon[/tex] for some [tex]N,\epsilon > 0[/tex]
Setting [tex]N=\sqrt{\frac{1-\epsilon}{\epsilon}}[/tex]
x > N
[tex]\Rightarrow[/tex] [tex] x > \sqrt{\frac{1-\epsilon}{\epsilon}}[/tex]
[tex]\Rightarrow[/tex] [tex] x^2 > \frac{1-\epsilon}{\epsilon}[/tex]
[tex]\Rightarrow[/tex] [tex] x^2 + 1 > \frac{1}{\epsilon}[/tex]
[tex]\Rightarrow[/tex] [tex] \frac{1}{1+x^2} < \epsilon [/tex]
[tex] \frac{1}{1+x^2} = f(x) [/tex]
and since N > 0, and x > N, it is implied x > 0 and therefore [tex]|f(x)| = f(x)[/tex]
I'm not sure if this is a good enough proof? Thanks in Advance :)