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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 2.6.80
Capitolo 2, Problema 2.6.80

Use the continuity of the absolute value function (Exercise 78) to determine the interval(s) on which the following functions are continuous.


g(x)=∣x+4x2−4∣g\(\left\)(x\(\right\))=\(\left\[\vert\]\frac{x+4}{x^2-4}\[\right\]\vert\)

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1
Identify the function: \( g(x) = \left| \frac{x+4}{x^2-4} \right| \).
Recognize that the absolute value function is continuous everywhere, so focus on the rational function \( \frac{x+4}{x^2-4} \).
Determine where the denominator \( x^2 - 4 \) is zero, as these points will be discontinuities. Solve \( x^2 - 4 = 0 \) to find \( x = 2 \) and \( x = -2 \).
Conclude that the function \( g(x) \) is continuous on the intervals where the denominator is not zero: \((-\infty, -2) \cup (-2, 2) \cup (2, \infty)\).
Verify that the function is not defined at \( x = 2 \) and \( x = -2 \), confirming these are points of discontinuity.

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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. Understanding continuity is essential for determining where the given function behaves without interruption.
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Intro to Continuity

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x, effectively removing any negative sign. This function is continuous everywhere on the real number line. When analyzing functions involving absolute values, it is crucial to consider how the absolute value affects the overall continuity and behavior of the function.
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Percorso guidato
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Initial Value Problems

Rational Functions and Their Discontinuities

A rational function is a ratio of two polynomials. Discontinuities in rational functions typically occur where the denominator equals zero, leading to undefined values. In the case of the function g(x) = |(x + 4)/(x^2 - 4)|, identifying the points where the denominator (x^2 - 4) is zero is vital for determining the intervals of continuity.
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Intro to Rational Functions
Pratica correlata
Domanda del libro di testo

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\)\(\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=12x4−4x8−9x4f\(\left\)(x\(\right\))=\(\frac{1}{2x^4-\sqrt{4x^8-9x^4}\)}

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Domanda del libro di testo

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→0 x^2=0 (Hint: Use the identity √x2=|x|.)

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Domanda del libro di testo

Determine the following limits at infinity.


lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

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Domanda del libro di testo

a. Analyze limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\)\(\lim\)_{x\(\to\)-\(\infty\)}{f(x)}} for each function.


f(x)=1+x−2x2−x3x2+1f\(\left\)(x\(\right\))=\(\frac{1+x-2x^2-x^3}{x^2+1}\)

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Domanda del libro di testo

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

f(x)=2x^2+3x+1 / x^2+5x; a=−5

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Domanda del libro di testo

Let f(x) =x^2−2x+3.


a. For ε=0.25, find the largest value of δ>0 satisfying the statement


|f(x)−2|<ε whenever 0<|x−1|<δ.

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