Question

Determine if each of the following statements is always true, sometimes true, or never true. Explain
a) If Lim(x-->0)f(x)=0, then f(x) is continuous at x=0

Answer 1

Sometimes true. Consider
\[f(x)=\frac{x^2}{x}\]
There's a hole at x = 0, so f(x) is not continuous at x = 0, despite the fact that
\[\lim_{x\to0}f(x)=0.\]

Answer 2

and what about this "If g(1)=3, then lim(x-->1)g(x)=3

Answer 3

Also sometimes true. Take the following function:
\[g(x)=\begin{cases}x&\text{if $x\not=1$}\\
3&\text{if $x=1$}\end{cases}\]
By this definition of g(x), you have g(1) = 3. However,
\[\lim_{x\to1}g(x)=\lim_{x\to1}x=1.\]