# Thread: Function with no limit at point c

1. ## Function with no limit at point c

Find a case of the function $\displaystyle f$ and $\displaystyle g$ where $\displaystyle f$ and $\displaystyle g$ do not have limits at a point c, but $\displaystyle f+g$ and $\displaystyle fg$ have limits at c.

Would

$\displaystyle f(x)=(-1)^x$
-and-
$\displaystyle g(x)=(-1)^{x+1}$

work? if $\displaystyle x>1$

2. What point c were you thinking of for your choice of f(x) and g(x)?

How about f(x) = 0 if x is rational; 1 if x is irrational. g(x) = 1 if x is rational; 0 if x is irrational. Then f(x) + g(x) = 1, f(x)g(x) = 0, and the limit does not exist for f(x) or for g(x) at any point c.

3. Just a general point C.

4. Don't complicate things.
$\displaystyle f(x) = \left\{ {\begin{array}{rl} 1 & {x > 0} \\ { - 1} & {x < 0} \\ \end{array} } \right.\quad \& \quad g(x) = \left\{ {\begin{array}{rl} { - 1} & {x > 0} \\ 1 & {x < 0} \\ \end{array} } \right.$

5. ok so there exists a limit at 3, but not at 0. I guess where I was getting confused is that I though c must be arbitrary so it had to go for all points.

6. Well O.K. Make it:
$\displaystyle f(x) = \left\{ {\begin{array}{rl} 1 & {x > c} \\ { - 1} & {x < c} \\ \end{array} } \right.\quad \& \quad g(x) = \left\{ {\begin{array}{rl} { - 1} & {x > c} \\ 1 & {x < c} \\ \end{array} } \right.$

Does that work? WHY or WHY NOT?