Question

Is the statement "if f(c)=L, then the limit of f(x)=L as x approaches c" a true or false statement?

Answer 1

It is not true in general. That is: There are situations in which `f(c)=L`, but it is not true that `lim_(xrarrc)f(x) = L`.

Explanation:

Example 1

Define `f(x) = {(1/x,"if",x != 0),(1,"if", x=0):}`

`f(0) = 1`, but `lim_(xrarr0)f(x)` does not exist.

Example 2

Define `g(x) = {(x,"if",x != 0),(1,"if", x = 0):}`

`g(0)=1`, but `lim_(xrarr0)g(x) = 0`