🎯 Limits Graph Lab
Skill goal: tell apart the limit of a function (where the graph is heading) from the function’s actual value at a point.
Observe
The graph of f(x) = (x² − 1)/(x − 1) looks like a straight line — but something is missing at x = 1. Drag the gold point along the curve and watch its coordinates.
Manipulate
👆 Drag the gold point left or right along the x-axis.
| x | f(x) |
|---|
Predict
Before you change it, what do you think will happen? Example: “If I drag the point closer and closer to x = 1, what value do the y-coordinates seem to approach?”
See
Explain — Why it works
A limit only cares about where the function is heading as x gets close to a point — not what happens exactly at that point. Here, (x² − 1)/(x − 1) = (x − 1)(x + 1)/(x − 1) = x + 1 for every x ≠ 1, so both sides march toward 2 even though f(1) itself is undefined (division by zero leaves a hole). The jump example shows the opposite: the left side heads to 1 and the right side to 3, so the two-sided limit does not exist.
Challenge
1. For f(x) = (x² − 1)/(x − 1), what is limx→1 f(x)? (Type a number.)
2. Switch to the jump example. What is limx→1 g(x) there?