Approaching a Point vs Being at the Point

This function follows the line y=x+1 everywhere near x=1, but at x=1 we deliberately set the function value to 3. The open circle shows the nearby pattern; the filled dot shows the actual value.

Learning goalDistinguish nearby behavior from the function value at the point.

f(x)=x+1 for x≠1·f(1)=3

Step 1

Move the idea

Drag the slider

Try it: compare nearby values with f(1)

Move the slider

Change one parameter and watch what changes with it.

Step 2 · The aha

The value at the point and the value approached nearby are two different pieces of information.

Filled dot

The filled dot at (1,3) answers the direct question: what is f(1)? The answer is 3.

Open circle

The open circle at (1,2) marks the value the surrounding line would have there. Nearby points from both sides get closer and closer to y=2.

Keep the questions separate

Function value asks what happens exactly at x=1. A limit asks what happens as x gets arbitrarily close to 1. They can agree, but they do not have to.

Step 3 · One-question check

Did it click?

For this graph, what is true at x=1?

Continue the path

Next: turn nearby behavior into the idea of a limit