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Limits Limits

Limits

A limit tells you what a function is approaching as x gets close to a number.

Think: “If I move closer and closer to x = a, what does y move closer to?”

lim as x → a of f(x) = L

Example 1: A hole (the limit exists)

The graph heads toward 2 near x = 1, even if the actual value at x = 1 is different.

x y -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 approaches 2 value is 3
Limit: 2

Example 2: A jump (no limit)

If the left side approaches a different number than the right side, the limit does not exist.

x y -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 x → 0− x → 0+
Left limit ≠ Right limit → no limit

Example 3: Blows up (infinite limit)

Near a vertical asymptote, values can shoot up or down.

x y -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 x = 1 x → 1+ x → 1−
As x → 1, the values go toward ±∞

Quick tip: A limit is about “getting close,” not necessarily “plugging in.”

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