Show a detailed solution for this one. Prove that if lim x => c of f(x) exists,
ID: 1890512 • Letter: S
Question
Show a detailed solution for this one.Prove that if lim x => c of f(x) exists, then f is bounded on some neighborhood of c.
(The function f: D => all real numbers R is bounded on some neighborhood of c if there exists a sigma > 0 and M > 0 such that(x-c) absolute value difference < sigma, x E D implies that abs(f(x) < M).
Explanation / Answer
False. A function does not have to be defined at a point to have a limit. Ex: a function with removeable discontinuity. Y= 2(x-3)/(x-3). Which is undefined at x= 3, but simplifies to y= 2 for all other values. So lim as x-> 3 is 2. It is just a horizontal line with a hole at (3,2) The limit is asking what do you expect y to be as you get closer to 3. You expect all the y's around it to be close to 2. But a function does have to be continuous to have a derivative at that point.