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Calculus/Proving limits

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Expert: - 5/16/2009


Question
Prove:  Limit as x approches infinity of 1/((x^2) + 1) =0

Please help. Thanks

Answer
Choose ε and take x > δ.

If ε = 1, to make 1/(x²+1) < 1, δ can be chosen as 0.
If x > 0, then 1/(x²+1) is less than ε.

If ε = 0.1, take δ = 3.  As can be seen, if x > δ, then the function is less than ε.

If ε = 0.01, take δ = 10.  That way if x > δ, the fuction is less than ε.

Take some ε however small.  It can be seen that with δ = 1/√ε,
this equation is true for all x > δ.

This proves that the limit is 0, for no matter how small of an ε is chosen, an appropriate δ can be chosen to make the function closer that that to 0.

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