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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 86
Capitolo 2, Problema 86

Use an appropriate limit definition to prove the following limits.


lim x→1 (5x−2) =3;

Guida verificata passo dopo passo
1
Start by recalling the definition of a limit: \( \lim_{{x \to a}} f(x) = L \) means that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).
Identify the function \( f(x) = 5x - 2 \), the point \( a = 1 \), and the limit \( L = 3 \).
Set up the inequality \( |f(x) - L| < \epsilon \) which becomes \( |(5x - 2) - 3| < \epsilon \). Simplify this to \( |5x - 5| < \epsilon \).
Factor the expression inside the absolute value: \( |5(x - 1)| < \epsilon \). This simplifies to \( 5|x - 1| < \epsilon \).
Solve for \( |x - 1| \) by dividing both sides by 5: \( |x - 1| < \frac{\epsilon}{5} \). Choose \( \delta = \frac{\epsilon}{5} \) to satisfy the limit definition.

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Limit Definition

The limit definition in calculus refers to the formal approach to determining the value that a function approaches as the input approaches a certain point. Specifically, for a function f(x), the limit as x approaches a value 'a' is L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε. This definition is foundational for proving limits rigorously.
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One-Sided Limits

Direct Substitution

Direct substitution is a method used to evaluate limits by substituting the value that x approaches directly into the function. If the function is continuous at that point, the limit can be found simply by replacing x with the target value. In the case of the limit lim x→1 (5x−2), substituting x = 1 yields the result 3, confirming the limit without further manipulation.
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Finding Limits by Direct Substitution

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 the limit lim x→1 (5x−2), the function is a polynomial, which is continuous everywhere. This property allows us to confidently use direct substitution to evaluate the limit, reinforcing the relationship between limits and continuity in calculus.
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Intro to Continuity
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