One-Sided Limits

Learn left-hand limits, right-hand limits.

Lesson 2 of 32 parts Practice Flashcards

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Left-Hand Limits

Objective: Explain left-hand limits, apply the stated method, and check its conditions.

The left-hand limit approaches a using inputs x<a. A piecewise rule’s branch valid immediately left of a determines this limit. Endpoint membership does not change which nearby branch is used, and a left limit can exist without a matching right limit. Approach direction chooses the neighboring rule.

Formulas

$limₓ→a⁻ f(x)$

Worked examples

Try itf(x)=6x−7 for x<7, and f(x)=6x+7 for x≥7. Find limₓ→7⁻ f(x).Show answer

Step 1: A left-hand approach uses the rule for x below the target. Step 2: Use 6x−7 and substitute 7 in its limiting expression. Step 3: The left-hand limit is 35.

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Left-Hand Limits

Objective: Explain left-hand limits, apply the stated method, and check its conditions.

Formulas

$limₓ→a⁻ f(x)$

Worked examples

Try itf(x)=6x−7 for x<7, and f(x)=6x+7 for x≥7. Find limₓ→7⁻ f(x).Show answer

Step 1: A left-hand approach uses the rule for x below the target. Step 2: Use 6x−7 and substitute 7 in its limiting expression. Step 3: The left-hand limit is 35.

Right-Hand Limits

Objective: Explain right-hand limits, apply the stated method, and check its conditions.

The right-hand limit approaches a using inputs x>a. Choose the rule valid immediately right of the target. A two-sided limit exists only when the left and right limits agree; an endpoint function value is a separate quantity. A right-hand limit alone does not establish a two-sided limit.

Formulas

$limₓ→a⁺ f(x)$

Worked examples

Try itf(x)=6x−7 for x<7, and f(x)=6x+7 for x≥7. Find limₓ→7⁺ f(x).Show answer

Step 1: A right-hand approach uses the rule for x above the target. Step 2: Use 6x+7. Step 3: The right-hand limit is 49.

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