Important Notes

• The limit exists if and only if both one-sided limits exist and are equal
• It doesn't matter if there's a hole or different value AT $x = a$
• Look at the trend of the curve, not individual points

Common Graph Scenarios

Scenario 1: Continuous Function

If the function is continuous at $a$, the limit equals the function value: $\lim_{x \to a} f(x) = f(a)$

Scenario 2: Removable Discontinuity (Hole)

Even if there's a hole at $x = a$, the limit can still exist. The limit is where the hole "should be" filled.

Scenario 3: Jump Discontinuity

If the function jumps from one value to another at $x = a$, the left and right limits differ, so the limit does not exist.

Scenario 4: Infinite Limit

If the function grows without bound as $x \to a$, we say the limit is infinite (or doesn't exist as a real number).

Practice Tip

Always check BOTH sides of the point. If they don't agree, the limit does not exist!

Assume the graph shows:

• Function approaches 3 from the left at $x=2$
• Function approaches 3 from the right at $x=2$
• There is a hole at $(2,3)$ and a solid dot at $(2,5)$