Below is a GeoGebra visualization you can use to explore the limit and continuity definitions.

The default function is $f(x) = 3x-2$. You can change this by entering a different expression using $x$ in the $f(x) = $ box.

The default value for $a$ is $1$. You can change this by moving the red dot on the domain (source) axis in the mapping diagram.

The default value for $L$ is $1.5$. You can change this by moving the blue diamond on the co-domain (target) axis in the mapping diagram.

The default value for $\epsilon$ is $0.5$. You can change this by moving the red dot on the $\epsilon$ slider in the mapping diagram or entering a value in the $\epsilon = $ box above the $\epsilon$ slider.

The default value for $\delta$ is $0.3$.You can change this by moving the red dot on the $\delta$ slider in the mapping diagram or entering a value in the $\delta = $ box above the $\delta$ slider.

To show (or hide) the basic definition of limit with $\epsilon$ and $\delta$ click on the box labeled "Definition of limit."

To show (or hide) samples for $|x-a| \le \delta$ click on the box labeled "Hide/show samples |x- a| ≤δ."

To show (or hide) samples for $|x-a| \le \delta$ click on the box labeled "Hide/show samples |x- a| ≤δ."

To show (or hide) the basic definition of
limit from above with $\epsilon$ and $\delta$ click on the box labeled
"Definition of limit from above."

To show (or hide) samples for $|x-a| \le \delta$ click on the box labeled "Hide/show samples 0< x- a ≤δ."

To show (or hide) samples for $|x-a| \le \delta$ click on the box labeled "Hide/show samples 0< x- a ≤δ."

To visualize continuity click on the box
labeled "For continuity show "x" and "f(x)" ". You can move the
blue dot on the domain (source)
axis in the mapping diagram to the desired $x$, and the appropriate
value of $f(x)$ will be determined and visualized on the co-domain
(target) axis in the mapping diagram. Adjust the blue diamond to match
the value for $f(a)$ and you are ready to explore the continuity of $f$
at $a$.

Notice how the graph is paired with the mapping diagram.

Martin Flashman, 20 Dec. 2014, Created with GeoGebra