Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 2 - Limits

Problema 2.7.25

Capitolo 2, Problema 2.7.25

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→7 f(x)=9, where f(x)={3x−12 if x≤7
x+2 if x>7

Guida verificata passo dopo passo

We need to prove that $ \lim_{{x \to 7}} f(x) = 9 $ using the precise definition of a limit. The function $ f(x) $ is defined piecewise: $ f(x) = 3x - 12 $ for $ x \leq 7 $ and $ f(x) = x + 2 $ for $ x > 7 $.

The precise definition of a limit states that for every $ \varepsilon > 0 $, there exists a $ \delta > 0 $ such that if $ 0 < |x - 7| < \delta $, then $ |f(x) - 9| < \varepsilon $.

For $ x \leq 7 $, $ f(x) = 3x - 12 $. We need $ |(3x - 12) - 9| < \varepsilon $. Simplifying, this becomes $ |3x - 21| < \varepsilon $, or $ |3(x - 7)| < \varepsilon $.

From $ |3(x - 7)| < \varepsilon $, we have $ |x - 7| < \frac{\varepsilon}{3} $. Thus, for $ x \leq 7 $, we can choose $ \delta = \frac{\varepsilon}{3} $.

7 \).> For $ x > 7 $, $ f(x) = x + 2 $. We need $ |(x + 2) - 9| < \varepsilon $. Simplifying, this becomes $ |x - 7| < \varepsilon $. Thus, for $ x > 7 $, we can choose $ \delta = \varepsilon $.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Definition of a Limit

The precise definition of a limit states that for a function f(x) to approach a limit L as x approaches a value c, for every ε > 0, there must exist a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This formal definition is crucial for proving limits rigorously.

Video consigliato:

05:50

One-Sided Limits

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, f(x) has two definitions: 3x - 12 for x ≤ 7 and x + 2 for x > 7. Understanding how to evaluate limits at points where the function changes its definition is essential for solving the limit problem.

Video consigliato:

Percorso guidato

05:36

Piecewise Functions

Epsilon-Delta Proof

An epsilon-delta proof involves demonstrating that for any chosen ε, a corresponding δ can be found that satisfies the limit definition. This method is used to show that as x approaches a certain value, the function's output can be made arbitrarily close to the limit, thus confirming the limit's existence.

Video consigliato:

3:56

Slope-Intercept Form

Pratica correlata

Domanda del libro di testo

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

$g (x) = ∣ \frac{x + 4}{x^{2} - 4} ∣$

639

views

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|.)

357

views

Domanda del libro di testo

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

237

views

Domanda del libro di testo

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$ , determine ${$\displaystyle$$\lim$_{x$\to$ a^{+}}}f$\left$(x$\right$)$ , ${$\displaystyle$$\lim$_{x$\to$ a^{-}}}f$\left$(x$\right$)$ , and ${$\displaystyle$$\lim$_{x$\to$ a}}f$\left$(x$\right$)$ .