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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.7.21
Chapter 2, Problem 2.7.21

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→4 x^2−16 / x−4=8 (Hint: Factor and simplify.)

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Step 1: Recognize that the expression \( \frac{x^2 - 16}{x - 4} \) is undefined at \( x = 4 \). To simplify, factor the numerator: \( x^2 - 16 = (x - 4)(x + 4) \).
Step 2: Simplify the expression by canceling the common factor \( x - 4 \) in the numerator and denominator, resulting in \( x + 4 \) for \( x \neq 4 \).
Step 3: According to the definition of a limit, for every \( \varepsilon > 0 \), there must exist a \( \delta > 0 \) such that if \( 0 < |x - 4| < \delta \), then \( \left| \frac{x^2 - 16}{x - 4} - 8 \right| < \varepsilon \).
Step 4: Substitute the simplified expression into the limit condition: \( \left| (x + 4) - 8 \right| = |x - 4| \).
Step 5: To satisfy the limit condition, choose \( \delta = \varepsilon \). This ensures that whenever \( 0 < |x - 4| < \delta \), \( |x - 4| < \varepsilon \), proving the limit exists and equals 8.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit Definition

The precise definition of a limit states that for a function f(x) to approach a limit L as x approaches a value a, for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This formalism is crucial for proving limits rigorously.
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Factoring and Simplifying

Factoring and simplifying expressions is a key technique in calculus, especially when evaluating limits. In the given limit, the expression x^2 - 16 can be factored as (x - 4)(x + 4), allowing for cancellation of the (x - 4) term in the denominator, which simplifies the limit evaluation.
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Epsilon-Delta Relationship

In the context of limits, the relationship between ε and δ is essential for proving that a limit exists. For the limit lim x→4 (x^2 - 16)/(x - 4) = 8, one must find a δ such that when x is within δ of 4, the value of the function is within ε of 8. This relationship ensures that the function behaves as expected near the limit point.
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Related Practice
Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 (8x+5)=13

Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→5 1/x^2=1/25

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Textbook Question

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=12x4−4x8−9x4f\(\left\)(x\(\right\))=\(\frac{1}{2x^4-\sqrt{4x^8-9x^4}\)}

Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→−3 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)

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Textbook Question

Determine the following limits at infinity.


lim x→−∞ 3x^11

Textbook Question

Determine the following limits at infinity.


lim t→∞ (−12t^−5)