Limit proofs
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Show that the following limits do not exist: use def. of convergence or cauchy or epsilon delta
a. lim as x goes to 0 of (1/(x^2)) x>0
b. lim as x goes to 0 of (1/(x^1/2)) same as (1/sqrtx)
c. lim as x goes to 0 of (x + sgn(x))
d. lim as x goes to 0 of sin(1/(x^2))
let f: R---> R be defined by setting f(x)= x if x is rational and f(x)=0 if x is irrational.
Show that if L=0 then lim as x goes to c of f(x) is 0
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Solution Summary
This solution provides examples of working with proofs of limits and their existence, and provides brief explanations.
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Problem #1
a. Let x_n = 1/n, then x_n goes to 0 as n goes to infinity. But we find that 1/x_n^2 = n^2 goes to infinity. So
the limit of 1/x^2 does not exist as x goes to 0.
b. Let x_n = 1/n^2, then x_n goes to 0 as n goes to infinity. But we find that 1/sqrt(x_n) = n goes to infinity. So
the limit of 1/sqrt(x) does not exist as x goes to 0.
c. Let x_n = ...
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