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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 2.4.6
Capitolo 2, Problema 2.4.6

Use the graph of f(x) = x / (x2 − 2x − 3)2 to determine lim x→−1 f(x) and lim x→3 f(x).

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1
Identify the function: \( f(x) = \frac{x}{(x^2 - 2x - 3)^2} \).
Factor the quadratic in the denominator: \( x^2 - 2x - 3 = (x - 3)(x + 1) \).
Rewrite the function: \( f(x) = \frac{x}{((x - 3)(x + 1))^2} \).
Analyze the behavior of \( f(x) \) as \( x \to -1 \): The denominator becomes zero, indicating a potential vertical asymptote or removable discontinuity.
Analyze the behavior of \( f(x) \) as \( x \to 3 \): Similarly, the denominator becomes zero, indicating another potential vertical asymptote or removable discontinuity.

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's value at points where it may not be explicitly defined, such as points of discontinuity or asymptotes. Evaluating limits often involves techniques like substitution, factoring, or applying L'Hôpital's rule.
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Continuity

Continuity refers to a property of a function where it is uninterrupted and has no breaks, jumps, or holes in its graph. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, especially when determining if a limit exists at a specific point.
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Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. They can be vertical, horizontal, or oblique and indicate the behavior of a function as it approaches certain values. Identifying asymptotes is crucial for understanding the limits of a function, particularly in cases where the function may tend toward infinity or exhibit undefined behavior at specific points.
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Use the precise definition of infinite limits to prove the following limits.


limx→41(x−4)2=∞{\(\displaystyle\)\(\lim\)_{x\(\to\)4}}\(\frac{1}{\left(x-4\right)^2}\)=\(\infty\)

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Determine the interval(s) on which the following functions are continuous. 

f(x)=x^5+6x+17 / x^2−9

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Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 

f(x)= √x−2; a=1

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Use the definitions given in Exercise 57 to prove the following infinite limits.


lim x→1^- 1 / 1 − x=∞

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Determine the following limits at infinity.


lim x→∞ (3+10/x^2)

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Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions.


g(2) =1,g(5) =−1,lim x→4 g(x) =−∞,lim x→7^− g(x) =∞,lim x→7^+ g(x) =−∞

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