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 2.49

Capitolo 2, Problema 2.49

Horizontal and Vertical Asymptotes

Determine the domain and range of y = (√16―x²) / (x―2).

Guida verificata passo dopo passo

Step 1: Identify the domain of the function. The domain is determined by the values of x for which the function is defined. The function y = (√(16 - x²)) / (x - 2) has two restrictions: the expression under the square root, 16 - x², must be non-negative, and the denominator, x - 2, must not be zero.

Step 2: Solve the inequality 16 - x² ≥ 0 to find the values of x for which the square root is defined. This inequality can be rewritten as x² ≤ 16, which implies -4 ≤ x ≤ 4.

Step 3: Determine the values of x that make the denominator zero. Set x - 2 = 0 and solve for x, which gives x = 2. Since the denominator cannot be zero, x = 2 is excluded from the domain.

Step 4: Combine the results from Steps 2 and 3 to find the domain. The domain is the set of all x such that -4 ≤ x ≤ 4, excluding x = 2. In interval notation, this is [-4, 2) ∪ (2, 4].

Step 5: Determine the range of the function. Consider the behavior of the function as x approaches the endpoints of the domain and the point where the denominator is zero. Analyze the limits as x approaches 2 from the left and right to identify any vertical asymptotes, and consider the maximum and minimum values of the function within the domain to determine the range.

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Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero. In this case, we need to identify any x-values that would cause the denominator (x - 2) to equal zero, as these values are excluded from the domain.

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Finding the Domain and Range of a Graph

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. To determine the range, we analyze the behavior of the function as x approaches certain critical points, including vertical asymptotes and the limits of the function as x approaches infinity or negative infinity. This helps in understanding the values that y can take.

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Asymptotes

Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur where the function is undefined, typically where the denominator is zero. Horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity, indicating the value that y approaches. Understanding these concepts is crucial for analyzing the overall behavior of the function.

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Introduction to Cotangent Graph

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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) = 1/x, x = -2

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Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0− h / sin 3h

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In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

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Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 2t / tan t

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Using the Formal Definition

Prove the limit statements in Exercises 37–50.