1. ## Limit Problem

Examples of functions satisfying each of the following conditions

(a) lim as x goes to zero of f of x^2 exists, but the lim as x goes to zero of f of x does not.

(b) lim as x goes to zero of f of 1/x exist, but the lim as x goes to zero of f of x does not.

(c) f(x) is not continuous at 1, but g(x) = |f(x)| is continuous at 1.

(d) f(x) is nto continuous anywhere, but g(x) = |f(x)| is continuous everywhere.

For each condition provide one function.

2. ## Re: Limit Problem

Originally Posted by calcmaster
Examples of functions satisfying each of the following conditions

(a) lim as x goes to zero of f of x^2 exists, but the lim as x goes to zero of f of x does not.

(b) lim as x goes to zero of f of 1/x exist, but the lim as x goes to zero of f of x does not.

(c) f(x) is not continuous at 1, but g(x) = |f(x)| is continuous at 1.

(d) f(x) is nto continuous anywhere, but g(x) = |f(x)| is continuous everywhere.
Here are two functions for you to think about.

$f(x) = \left\{ {\begin{array}{rl} {1,}&{x \geqslant 0} \\ { - 1,}&{x < 0} \end{array}} \right.$ AND $g(x)=\frac{1}{x}\text{ if }x\ne 0,~\&~0\text{ otherwise}.$

3. ## Re: Limit Problem

Another interesting function would be f(x)= 1 if x is irrational, f(x)= -1 if x is rational.