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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.R.79
Capitolo 2, Problema 2.R.79

Let g(x)={5x−2if x<1aif x=1ax2+bxif x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)5x-2 & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ ax^2+bx & \(\text{if }\)x>1\(\end{cases}\).


Determine values of the constants aa and bb , if possible, for which gg is continuous at x=1x=1.

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To ensure the function g(x) is continuous at x = 1, the left-hand limit, right-hand limit, and the value of the function at x = 1 must all be equal.
First, calculate the left-hand limit as x approaches 1. For x < 1, g(x) = 5x - 2. Thus, the left-hand limit is lim_{x \(\to\) 1^-} g(x) = 5(1) - 2 = 3.
Next, calculate the right-hand limit as x approaches 1. For x > 1, g(x) = ax^2 + bx. Thus, the right-hand limit is lim_{x \(\to\) 1^+} g(x) = a(1)^2 + b(1) = a + b.
The function value at x = 1 is given by g(1) = a.
For g(x) to be continuous at x = 1, set the left-hand limit equal to the right-hand limit and the function value: 3 = a and 3 = a + b. Solve these equations to find the values of a and b.

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Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function g(x) has three distinct cases depending on whether x is less than, equal to, or greater than 1. Understanding how to evaluate and analyze piecewise functions is crucial for determining continuity and limits at specific points.
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Piecewise Functions

Continuity

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 g(x) to be continuous at x=1, the left-hand limit (as x approaches 1 from the left) must equal the right-hand limit (as x approaches 1 from the right) and also equal g(1). This concept is essential for solving the problem of finding appropriate values for a and b.
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Intro to Continuity

Limits

Limits describe the behavior of a function as it approaches a certain point. In this context, we need to calculate the limits of g(x) as x approaches 1 from both sides. The values of a and b can be determined by ensuring that these limits match the value of g(1), which is defined as a, thus ensuring the function is continuous at that point.
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One-Sided Limits
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