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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 2.50
2장, 문제 2.50

Horizontal and Vertical Asymptotes
Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .

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To find the vertical asymptotes, set the denominator equal to zero and solve for x. This is because vertical asymptotes occur where the function is undefined. So, solve the equation x - b = 0.
The solution to x - b = 0 is x = b. Therefore, there is a vertical asymptote at x = b.
To find the horizontal asymptotes, analyze the behavior of the function as x approaches infinity or negative infinity. Consider the leading terms in the numerator and denominator.
The leading term in the numerator is √(ax²), which simplifies to √a * x. The leading term in the denominator is x. As x approaches infinity, the function behaves like (√a * x) / x.
Simplify (√a * x) / x to √a. Therefore, the horizontal asymptote is y = √a.

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주요 개념

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Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input approaches infinity or negative infinity. For rational functions, they can often be determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If they are equal, the asymptote is the ratio of the leading coefficients.
추천 영상:
5:46
Graphs of Exponential Functions

Vertical Asymptotes

Vertical asymptotes occur where a function approaches infinity, typically at values of x that make the denominator zero while the numerator is non-zero. To find vertical asymptotes, set the denominator equal to zero and solve for x. These points indicate where the function is undefined and can lead to infinite behavior in the graph.
추천 영상:
3:40
Introduction to Cotangent Graph Example 1

Rational Functions

A rational function is a function represented by the ratio of two polynomials. The behavior of rational functions, particularly their asymptotic behavior, is influenced by the degrees and coefficients of the polynomials in the numerator and denominator. Understanding the structure of rational functions is essential for analyzing their graphs, including identifying asymptotes.
추천 영상:
6:04
Intro to Rational Functions
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Using the Formal Definition


Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.


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Slope of a Curve at a Point


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y=7−x², P(2,3)

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Using the Sandwich Theorem


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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→∞ √(x² + 1) / (x + 1)

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