Question

Is the statement "if f is undefined at x=c, then the limit of f(x) as x approaches c does not exist" a true or false statement?

Answer 1

It is false. (That is, we cannot infer from "`f` is undefined at `c`" that the limit fails to exist.)

Explanation:

Counterexamples

`f(x) = xsin (1/x)` is undefined at `0`, but `lim_(xrarr0)xsin(1/x) = 0`

`g(x) = (x^2-1)/(x^2-x)` is undefined at `x=1`,

but `lim_(xrarr1) (x^2-1)/(x^2-x) = lim_(xrarr1) (x+1)/x = 2`