Domanda del libro di testo
Use the precise definition of infinite limits to prove the following limits.
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Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 2.6.27
Determine the interval(s) on which the following functions are continuous.
f(x)=x^5+6x+17 / x^2−9
Guida verificata passo dopo passo1
Identify the type of function: \( f(x) = \frac{x^5 + 6x + 17}{x^2 - 9} \) is a rational function, which is continuous everywhere except where the denominator is zero.
Set the denominator equal to zero to find the points of discontinuity: \( x^2 - 9 = 0 \).
Solve the equation \( x^2 - 9 = 0 \) to find the values of \( x \) that make the denominator zero. This can be factored as \( (x - 3)(x + 3) = 0 \).
Determine the solutions to the equation: \( x = 3 \) and \( x = -3 \). These are the points where the function is not continuous.
Conclude that the function \( f(x) \) is continuous on the intervals \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).
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Continuity of Functions
A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. This concept is crucial for determining where a function does not have breaks, jumps, or asymptotes.
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05:34
Intro to Continuity
Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials. The continuity of rational functions is affected by the values that make the denominator zero, as these points create vertical asymptotes or discontinuities. Understanding the behavior of rational functions helps in identifying intervals of continuity.
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6:04Intro to Rational Functions
Finding Intervals of Continuity
To find the intervals of continuity for a function, one must identify points where the function is undefined, typically where the denominator is zero. After determining these points, the intervals can be established by testing the function's behavior in the regions between these points. This process allows for a comprehensive understanding of where the function remains continuous.
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03:38Intro to Continuity Example 1
Pratica correlata
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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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Domanda del libro di testo
Use the definitions given in Exercise 57 to prove the following infinite limits.
lim x→1^- 1 / 1 − x=∞
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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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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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Domanda del libro di testo
A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist.
{0,1/2,2/3,3/4,…}, which is defined by f(n) = (n−1) / n, for n=1,2,3,…
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