Limits from Graphs and Tables

Learn graphical limit evidence, table evidence and a known rule, limits at holes.

Lesson 2 of 33 parts Practice Flashcards

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Graphical Limit Evidence

Objective: Explain graphical limit evidence, apply the stated method, and check its conditions.

A graph’s nearby behavior determines a limit, while a filled point determines its assigned value. A two-sided finite limit requires matching approach heights. A graph description must state both sides; a single plotted point alone cannot establish the surrounding limit. A filled point shows a value, not necessarily the limit.

Formulas

Two-sided limit requires Lleft=Lright

Worked examples

Try itA described graph approaches height 42 from both sides as x approaches 7, but its filled point at x=7 has height 49. Find the two-sided limit.Show answer

Step 1: Both approach directions have the same stated height. Step 2: That limiting height is 42, regardless of the filled point. Step 3: The limit is 42; the function value is 49.

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Graphical Limit Evidence

Objective: Explain graphical limit evidence, apply the stated method, and check its conditions.

Formulas

Two-sided limit requires Lleft=Lright

Worked examples

Try itA described graph approaches height 42 from both sides as x approaches 7, but its filled point at x=7 has height 49. Find the two-sided limit.Show answer

Step 1: Both approach directions have the same stated height. Step 2: That limiting height is 42, regardless of the filled point. Step 3: The limit is 42; the function value is 49.

Table Evidence and a Known Rule

Objective: Explain table evidence and a known rule, apply the stated method, and check its conditions.

Values sampled increasingly near a target can suggest a limit, but finitely many entries do not prove arbitrary nearby behavior. If the table is generated by a stated continuous rule, continuity justifies substituting the target. Separate observed evidence from the supporting rule. A finite table alone cannot prove every limit claim.

Formulas

For continuous f: limₓ→a f(x)=f(a)

Worked examples

Try itA synthetic table is generated by the continuous rule f(x)=6x+7, with inputs approaching 7 from both sides. Find the limiting output.Show answer

Step 1: The stated rule is continuous at the target input. Step 2: Evaluate 6(7)+7. Step 3: The limit is 49; finitely many table entries alone would not prove a limit without the given rule.

Limits at Holes

Objective: Explain limits at holes, apply the stated method, and check its conditions.

A 0/0 substitution gives an indeterminate form. Factoring may expose a removable factor and an equivalent nearby rule. Cancel only away from the target, then evaluate the simplified rule’s limit; the original function may still be undefined at the target. 0/0 is indeterminate, not an answer of zero.

Formulas

$(x²−a²)/(x−a)=x+a for x≠a$

Worked examples

Try itFind limₓ→7 (x²−49)/(x−7).Show answer

Step 1: Factor numerator: (x−7)(x+7). Step 2: For x≠7, cancel to get x+7. Step 3: The limit is 14.

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