Limits Of Exotic Functions

Calculus Level 4

\[ \begin{align} f(x) &= \begin{cases} 0 &\text{ if } x \text{ is irrational} \\ 1 &\text{ if } x \text{ is rational} \end{cases} \\ g(x) &= \begin{cases} 0 &\text{ if } x \text{ is irrational} \\ \frac1q &\text{ if } x =\frac{p}{q}, \text{ where } p \text{ and } q \text{ are coprime nonnegative integers} \end{cases} \end{align} \]

Let \( f(x) \) and \( g(x) \) be two functions defined on \( [0,1] \) by the formulas as described above.

For which \( a \in (0,1) \) does \( \lim\limits_{x\to a} f(x) \) exist?

For which \( b \in (0,1) \) does \( \lim\limits_{x\to b} g(x) \) exist?

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