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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 2.6.91
Chapter 2, Problem 2.6.91

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.

Verified step by step guidance
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Step 1: Understand the formal definition of a limit as x approaches infinity. The limit of a function f(x) as x approaches infinity is the value that f(x) gets closer to as x becomes very large.
Step 2: Recognize that if f(x) is a constant function, meaning f(x) = k for all x, then as x approaches infinity, f(x) remains constant at k.
Step 3: Apply the formal definition of limits. For any ε > 0, there exists a number N such that for all x > N, the absolute difference |f(x) - k| is less than ε.
Step 4: Since f(x) = k for all x, the absolute difference |f(x) - k| is 0, which is always less than any positive ε. Therefore, the condition of the limit definition is satisfied.
Step 5: Conclude that the limit of f(x) as x approaches infinity is k, because the function value does not change and remains equal to k for all x.

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Key Concepts

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

Limit of a Function

The limit of a function describes the behavior of that function as the input approaches a certain value. In the context of limits as x approaches infinity, it examines how the function behaves as x grows larger and larger. Understanding limits is fundamental in calculus, as it lays the groundwork for concepts such as continuity, derivatives, and integrals.
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Formal Definition of a Limit

The formal definition of a limit, often expressed using epsilon-delta notation, provides a rigorous way to define what it means for a function to approach a certain value as the input approaches a specific point. For limits at infinity, this definition helps establish that for every small positive number (epsilon), there exists a corresponding value of x (N) such that for all x greater than N, the function's value is within epsilon of the limit.
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Constant Function Limits

A constant function is one where the output value remains the same regardless of the input, expressed as f(x) = k. The limit of a constant function as x approaches infinity is simply the constant value itself, lim x → ∞ f(x) = k. This concept is crucial for understanding how functions behave at extreme values and simplifies the analysis of limits in calculus.
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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))

Textbook Question

Exercises 5–10 refer to the function

f(x) = { x² − 1, −1 ≤ x < 0

2x, 0 < x < 1

1, x = 1

−2x + 4, 1 < x < 2

0, 2 < x < 3

graphed in the accompanying figure.

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At what values of x is f continuous?

Textbook Question

Use formal definitions to prove the limit statements in Exercises 93–96.


lim x → 0 (1 / |x|) = ∞

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Textbook Question

Graphing Simple Rational Functions


Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.


y = 1/(x − 1)

Textbook Question

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

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