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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 2 - Limits and Continuity문제 2.6.71
2장, 문제 2.6.71

Domains and Asymptotes


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


y = (√(x² + 4)) / x

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Step 1: Determine the domain of the function y = (√(x² + 4)) / x. The domain consists of all x-values for which the function is defined. Since the square root function √(x² + 4) is defined for all real numbers, we need to ensure the denominator x is not zero to avoid division by zero. Therefore, the domain is all real numbers except x = 0.
Step 2: Identify vertical asymptotes by examining the behavior of the function as x approaches values that are not in the domain. Since x = 0 is not in the domain, check the limit of y as x approaches 0 from the left and right. If the limit approaches infinity or negative infinity, x = 0 is a vertical asymptote.
Step 3: Determine horizontal asymptotes by evaluating the limit of y as x approaches positive and negative infinity. Calculate lim(x→∞) (√(x² + 4)) / x and lim(x→-∞) (√(x² + 4)) / x. Simplify the expression by dividing the numerator and the denominator by x, which is the highest power of x in the denominator.
Step 4: Simplify the expression (√(x² + 4)) / x by dividing both the numerator and the denominator by x. This results in √(1 + 4/x²). As x approaches infinity, 4/x² approaches 0, so the expression simplifies to √1 = 1. Therefore, the horizontal asymptote is y = 1.
Step 5: Conclude the analysis by summarizing the domain and asymptotes. The domain of the function is all real numbers except x = 0. The function has a vertical asymptote at x = 0 and a horizontal asymptote at y = 1.

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Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function y = (√(x² + 4)) / x, the domain excludes x = 0 because division by zero is undefined. Additionally, the expression under the square root, x² + 4, must be non-negative, which is always true for real numbers.
추천 영상:
5:10
Finding the Domain and Range of a Graph

Limits and Asymptotes

Limits help determine the behavior of a function as the input approaches a particular value, which is crucial for identifying asymptotes. Vertical asymptotes occur where the function becomes undefined, often where the denominator is zero. Horizontal or oblique asymptotes are found by evaluating the limit of the function as x approaches infinity or negative infinity.
추천 영상:
03:07
Cases Where Limits Do Not Exist

Square Root Function

The square root function, √(x), is defined for non-negative values of x. In the context of y = (√(x² + 4)) / x, the expression x² + 4 is always positive, ensuring the square root is defined for all real x. Understanding how the square root affects the function's behavior is essential for analyzing its domain and asymptotic properties.
추천 영상:
03:19
Completing the Square to Rewrite the Integrand Example 6
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교과서 질문

Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

x³ − 3x − 1 = 0

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교과서 질문

Use formal definitions to prove the limit statements in Exercises 93–96.


lim x → 0 (−1 / x²) = −∞

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교과서 질문

Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


lim x→0 (−1) / (x² (x + 1))

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교과서 질문

Graphing Simple Rational Functions


Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.


y = −3/(x − 3)

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Limits with trigonometric functions


Find the limits in Exercises 43–50.


limx→0 (1 + x + sin x) / (3 cosx)

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Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=x³−3x²+4, P(2,0)

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