Concept preview

What you will be able to do

1. 01State informally what $\lim_{x\to c} f(x)=L$ means in terms of outputs approaching $L$ as inputs approach $c$.
2. 02Use and read one-sided limit notation, and connect the two-sided limit to its one-sided parts.
3. 03Explain why the value the function approaches at $c$ is independent of the value (if any) the function takes at $c$.

Imagine walking toward a doorway. The limit answers “where are you heading?”, not “are you standing in the doorway?”. In symbols we write $\lim_{x\to c} f(x)=L,$ read “the limit of $f(x)$ as $x$ approaches $c$ equals $L$”. It means: we can make the outputs $f(x)$ as close to $L$ as we like, simply by taking $x$ close enough to $c$ (but not equal to $c$). The phrase in parentheses is the whole point — the input creeps toward $c$ without ever being required to land on it.