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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 2.6.87
Capitolo 2, Problema 2.6.87

Let g(x)={x2+xif x<1aif x=13x+5if x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)
a. Determine the value of a for which gg is continuous from the left at 11.

Guida verificata passo dopo passo
1
To determine the value of 'a' for which the function g(x) is continuous from the left at x = 1, we need to ensure that the left-hand limit as x approaches 1 is equal to g(1).
First, find the left-hand limit of g(x) as x approaches 1. Since g(x) = x^2 + x for x < 1, we calculate the limit of x^2 + x as x approaches 1 from the left.
Evaluate the expression x^2 + x at x = 1. This gives us the left-hand limit: lim_{x \(\to\) 1^-} (x^2 + x).
For g(x) to be continuous from the left at x = 1, this left-hand limit must equal g(1), which is 'a'. Therefore, set the left-hand limit equal to 'a'.
Solve the equation obtained in the previous step to find the value of 'a' that makes g(x) continuous from the left 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 from both sides equals the function's value at that point. For piecewise functions, this often involves ensuring that the left-hand limit and right-hand limit at the point of interest are equal to the function's value at that point.
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Intro to Continuity

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. In this case, the function g(x) has three different definitions depending on whether x is less than, equal to, or greater than 1. Understanding how to evaluate these expressions is crucial for analyzing the function's behavior at specific points.
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Piecewise Functions

Limits

Limits describe the behavior of a function as it approaches a certain point. In the context of continuity, we need to evaluate the left-hand limit (as x approaches 1 from the left) and the right-hand limit (as x approaches 1 from the right) to determine if they are equal and match the function's value at that point, which is essential for ensuring continuity.
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One-Sided Limits
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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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Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

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Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→3 −5x / √4x − 3

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

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Determine the following limits.


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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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