F(x) = x if x is rational, 0 if x is irrational Use the δ, ε definition of the limit to prove that lim(x→0)f(x)=0 Use the δ, ε definition of the limit to prove that lim(x→a)f(x) does not exist
F x 0 if x is rational 1 if x is irrational graph-My thoughts so far (1) Its graph seems to show this function acting as though it were twoGraph of the irrational function f ( x) = x 2 4 Again, we have to start by finding the domain of the function To do this, we have to solve the quadratic that is inside the square root, but we can also
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This is a discontinuous function, the graph would be just "dots" at either y = 0 or at y = 1 This is an "even" function, because the negative of any positive rational would have a corresponding y Suppose lim x → 0 g ( x) exists with limit L Then for every ϵ > 0, there exists a δ > 0 such that for all x, x − 1 < δ implies g ( x) − L < ϵ Choose ϵ < 1 / 2 If L = 0, then taking x = 1
Incoming Term: f x 0 if x is rational 1 if x is irrational graph,
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