Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 2.6.27

Capitolo 2, Problema 2.6.27

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→∞ (2√x + x⁻¹) / (3x − 7)

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

Divide every term in the numerator and the denominator by x, the highest power of x in the denominator.

Rewrite the expression: (2√x/x + x⁻¹/x) / (3x/x - 7/x).

Simplify each term: (2/√x + 1/x²) / (3 - 7/x).

Evaluate the limit as x approaches infinity: As x → ∞, 2/√x → 0, 1/x² → 0, and 7/x → 0, so the limit simplifies to 0/3.

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Limits at Infinity

Limits at infinity refer to the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, particularly for rational functions where the degree of the numerator and denominator can determine the limit. Analyzing limits at infinity helps in identifying horizontal asymptotes and the end behavior of functions.

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Cases Where Limits Do Not Exist

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The degree of the polynomials in the numerator and denominator plays a significant role in determining the limit as x approaches infinity. Understanding the structure of rational functions allows for the simplification of limits by dividing through by the highest power of x, which reveals the dominant terms that dictate the limit's value.

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Intro to Rational Functions

Dominant Terms

In the context of limits, dominant terms are the terms in a polynomial that have the highest degree and thus have the most significant impact on the function's behavior as x approaches infinity. When evaluating limits, identifying and focusing on these terms allows for simplification of the expression, making it easier to determine the limit. This concept is essential for effectively applying the technique of dividing by the highest power of x.

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Simplifying Trig Expressions Example 1

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Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = x², x = -2

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Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 2x / (x² − 1)

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If functions f(x) and g(x) are continuous for 0 ≤ x ≤ 1, could f(x)/g(x) possibly be discontinuous at a point of [0,1]? Give reasons for your answer.

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