1. ## Limits

Hi I am having problems with a couple of questions on my current assignment. I assume both questions are relating to the delta/epsilon proof of limits, which was given alongside the assignment.

Lets f1(x) = x, f2(x) = 2x, ..., fn(x) = nx. Then lim[x->0] fn(x)=0 for each n. Does there exist a positive number delta such that for every positive integer n we have absolute(fn(x)-0) <1 whenever 0< absolute(x-0) < delta

Let f(x) = 1 if x is rational, 0 if x is irrational. Show that lim[x->0] f(x) does not exist.

2. Originally Posted by Webby
Hi I am having problems with a couple of questions on my current assignment. I assume both questions are relating to the delta/epsilon proof of limits, which was given alongside the assignment.

Lets f1(x) = x, f2(x) = 2x, ..., fn(x) = nx. Then lim[x->0] fn(x)=0 for each n. Does there exist a positive number delta such that for every positive integer n we have absolute(fn(x)-0) <1 whenever 0< absolute(x-0) < delta

Let f(x) = 1 if x is rational, 0 if x is irrational. Show that lim[x->0] f(x) does not exist.

1. No

For any $\delta > 0$, we can find a natural number N such that $x^* = 1/N < \delta$ but $f_N(x^*) = N*\frac{1}{N} = 1$

2. Hint: look at $\epsilon < 1$ and use the fact that rational numbers are dense on any interval to show that the definition of continuity will not hold.

3. Originally Posted by Webby

Let f(x) = 1 if x is rational, 0 if x is irrational. Show that lim[x->0] f(x) does not exist.
A very useful nugget of information to have locked up in your noodle is that since $\mathbb{Q},\mathbb{R}-\mathbb{Q}$ are dense in $\mathbb{R}$ for each $x\in\mathbb{R}$ there are sequences $\{q_n\},\{i_n\}$ in each such that $\lim\text{ }q_n=x=\lim\text{ }i_n$.