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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 2.6.39
Capitolo 2, Problema 2.6.39

Complete the following steps for each function.


c. State the interval(s) of continuity.


f(x)={2x if x<1
x^2+3x if x≥1; a=1

Guida verificata passo dopo passo
1
Step 1: Understand the piecewise function. The function f(x) is defined as two separate expressions depending on the value of x. For x < 1, f(x) = 2x. For x \(\geq\) 1, f(x) = x^2 + 3x.
Step 2: Determine the continuity at the point where the function changes, which is at x = 1. For a function to be continuous at a point, the left-hand limit, right-hand limit, and the function value at that point must all be equal.
Step 3: Calculate the left-hand limit as x approaches 1 from the left (x \(\to\) 1^-). Use the expression for x < 1, which is 2x. Evaluate the limit: \(\lim\)_{{x \(\to\) 1^-}} 2x.
Step 4: Calculate the right-hand limit as x approaches 1 from the right (x \(\to\) 1^+). Use the expression for x \(\geq\) 1, which is x^2 + 3x. Evaluate the limit: \(\lim\)_{{x \(\to\) 1^+}} (x^2 + 3x).
Step 5: Check the value of the function at x = 1. Since x = 1 falls into the second piece of the function (x \(\geq\) 1), calculate f(1) using x^2 + 3x. Compare this value with the left-hand and right-hand limits to determine if the function is continuous at x = 1.

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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 a function behaves predictably without breaks, jumps, or asymptotes.
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Intro to Continuity

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has two distinct expressions: one for x < 1 and another for x ≥ 1. Analyzing piecewise functions requires checking the continuity at the boundaries where the pieces meet, which is crucial for determining the overall continuity of the function.
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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. To assess continuity at the boundary of a piecewise function, one must evaluate the left-hand limit and the right-hand limit at that point. If both limits exist and are equal to the function's value at that point, the function is continuous there.
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