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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 2.6.7
Chapter 2, Problem 2.6.7

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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Identify the dominant terms in the numerator and the denominator as x approaches infinity. For h(x) = \( \frac{-5 + \frac{7}{x}}{3 - \frac{1}{x^2}} \), the dominant terms are -5 in the numerator and 3 in the denominator.
As x approaches infinity, the terms \( \frac{7}{x} \) and \( \frac{1}{x^2} \) approach zero. Therefore, the expression simplifies to \( \frac{-5}{3} \).
Thus, the limit of h(x) as x approaches infinity is \( \frac{-5}{3} \).
Now, consider the limit as x approaches negative infinity. Again, the terms \( \frac{7}{x} \) and \( \frac{1}{x^2} \) approach zero, simplifying the expression to \( \frac{-5}{3} \).
Therefore, the limit of h(x) as x approaches negative infinity is also \( \frac{-5}{3} \).

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Key Concepts

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

Limits at Infinity

Limits at infinity involve determining the behavior of a function as the input approaches positive or negative infinity. This concept helps in understanding the end behavior of functions, often simplifying expressions by focusing on dominant terms. For rational functions, this typically involves comparing the degrees of the numerator and denominator.
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Cases Where Limits Do Not Exist

Simplification of Rational Functions

Simplification of rational functions is crucial when finding limits at infinity. It involves identifying and focusing on the dominant terms in the numerator and denominator, as terms with higher powers of x will dictate the behavior of the function as x approaches infinity. This simplification often leads to easier computation of limits.
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Intro to Rational Functions

Graphical Visualization

Graphical visualization aids in understanding the behavior of functions as x approaches infinity or negative infinity. By using graphing tools, one can observe the asymptotic behavior and confirm analytical results. This visual approach complements algebraic methods, providing a clearer picture of the function's end behavior.
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Determining Differentiability Graphically
Related Practice
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

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.

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

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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