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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
1
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))

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Textbook Question

Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → ∞ (√(x² + x) − √(x² − x))

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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?

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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)

348
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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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Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→9 √(x − 5) = 2

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