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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Identify the highest power of x in the denominator, which is x² in this case.

Divide both the numerator and the denominator by x², the highest power of x in the denominator.

Rewrite the expression: ((1/x² - x³/x²) / (1 + 7/x))⁵.

Simplify the expression: ((1/x² - x) / (1 + 7/x))⁵.

Evaluate the limit as x approaches -∞. As x → -∞, 1/x² → 0, 7/x → 0, and -x → ∞. Therefore, the expression simplifies to ((0 - ∞) / (1 + 0))⁵, which further simplifies to (-∞)⁵.

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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 finding the behavior of a function as the variable approaches positive or negative infinity. This concept helps determine the end behavior of functions, particularly rational functions, by analyzing the dominant terms that influence the function's growth or decay.

Rational Functions

Rational functions are quotients of two polynomials. Understanding their limits involves identifying the highest power of x in the denominator and numerator, which dictates the function's behavior as x approaches infinity. Simplifying by dividing by the highest power of x helps reveal the dominant terms affecting the limit.

Dominant Terms

Dominant terms in a polynomial are those with the highest degree, which significantly influence the function's behavior as x approaches infinity. By focusing on these terms, we can simplify the limit calculation, as lower-degree terms become negligible, allowing us to determine the limit more efficiently.

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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

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)

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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.