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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.3.39
Chapter 2, Problem 2.3.39

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→9 √(x − 5) = 2

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1
Step 1: Understand the formal definition of a limit. The limit of a function f(x) as x approaches a value c is L if for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.
Step 2: Identify the function and the limit statement. Here, the function is f(x) = √(x - 5) and we want to prove that lim(x→9) f(x) = 2.
Step 3: Set up the inequality |√(x - 5) - 2| < ε. We need to find a δ such that whenever 0 < |x - 9| < δ, this inequality holds.
Step 4: Manipulate the inequality |√(x - 5) - 2| < ε to find a suitable δ. Start by squaring both sides to eliminate the square root, leading to |x - 5 - 4| < ε².
Step 5: Solve the inequality |x - 9| < δ by relating it to the previous inequality. Choose δ such that it satisfies both the original limit condition and the manipulated inequality, ensuring the limit holds as x approaches 9.

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Key Concepts

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

Limit Definition

The formal definition of a limit, often called the epsilon-delta definition, states that for a function f(x) to have a limit L as x approaches a value c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This definition is crucial for rigorously proving limit statements.
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Definition of the Definite Integral

Square Root Function

Understanding the behavior of the square root function is essential, as it is continuous and differentiable for x > 0. In this context, the function √(x - 5) is involved, and knowing its properties helps in manipulating and evaluating the limit as x approaches a specific value, ensuring the function behaves predictably.
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Completing the Square to Rewrite the Integrand Example 6

Substitution Method

The substitution method is a technique used to simplify limit problems by introducing a new variable. For the given limit, setting u = x - 5 transforms the problem into a simpler form, allowing us to apply the epsilon-delta definition more easily. This method helps in isolating the variable and focusing on the core behavior of the function near the limit point.
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Related Practice
Textbook Question

Finding Limits


In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)


h(x) = (−5 + (7/x))/(3 – (1/x²))

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Textbook Question

Using the Formal Definitions


Use the formal definitions of limits as x → ±∞ to establish the limits in Exercises 91 and 92.


If f has the constant value f(x) = k, then lim x → ∞ f(x) = k.

315
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Textbook Question

Define h(2) in a way that extends h(t) = (t² + 3t − 10)/(t − 2) to be continuous at t = 2.

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Textbook Question

Using Limit Rules


Suppose lim x→0 f(x) = 1 and lim x→0 g(x) = −5. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.


limx→0 (2f(x) − g(x)) / (f(x) + 7)² = limx→0 (2f(x) − g(x)) / limx→0 (f(x) + 7)² (a)


(We assume the denominator is nonzero.)


(lim x→0 2f(x) − lim x→0 g(x)) / (lim x→0 (f(x) + 7))² (b)


= (2 lim x→0 f(x) − lim x→0 g(x)) / (lim x→0 f(x) + lim x→0 7)² (c)


= ((2)(1) − (−5)) / (1 + 7)² = 7/64

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Textbook Question

For what values of a and b is

g(x) = { ax + 2b, x ≤ 0

x² + 3a – b, 0 < x ≤ 2

3x – 5, x > 2

continuous at every x?

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Textbook Question

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x → ⁻∞ ((1 − x³) / (x² + 7x))⁵

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