# Limits 1

This worksheet illustrates the formal ($\epsilon{-}\delta$) definition of the limit of a real-valued function.

[list=1] [*] Move the point [color=#0000ff]$a$[/color] and adjust $\epsilon$. [*] The [color=#ff0000]red[/color] line on the $y$-axis denotes the minimum and maximum values attained by $f$ over the interval $(a-\delta,a+\delta)$. [*] Vary $\delta$ and observe that no matter how small $\epsilon$, you can always set $\delta$ so that $|f(x)-l|<\epsilon$. That is, the red line can always be shrunk so that it fits between the upper and lower green dots. [/list]