Prerequisites
Read coordinates, evaluate a linear rule and distinguish inputs smaller or larger than a target.
For a two-sided finite limit at x = a, nearby outputs from both sides must approach the same number. The value at a is a separate question. A different value or a missing value at that single input need not change the limit. Use the stated rules, not just a few table rows.
Worked example
A separate function H has H(x) = x - 1 whenever x is not 4, but H(4) = 8. On either side of 4, the nearby rule approaches 3. Thus the limit is 3 while the point value is 8. Setting H(4) to 3 would make H continuous there.
Three function records
A: For every x other than 2, A(x) = x + 4. At exactly 2, A(2) = -1. The graph shows this rule, an open circle at (2, 6), and the filled point (2, -1). B: For x < 3, B(x) = 2. For x > 3, B(x) = 7. At exactly 3, B(3) = 4. C: Only four readings are known: C(0.9) = 2.9, C(0.99) = 2.99, C(1.01) = 3.01, C(1.1) = 3.1. No rule for other nearby inputs is supplied.