Skip to main content
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 53b
Capitolo 2, Problema 53b

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.
lim x→−2^− f(x)

Guida verificata passo dopo passo
1
Identify the property of even functions: An even function satisfies the condition f(-x) = f(x) for all x in its domain.
Given that f is even, we have f(-x) = f(x). Therefore, f(-2) = f(2).
Examine the given limits: \( \lim_{x \to 2^+} f(x) = 5 \) and \( \lim_{x \to 2^-} f(x) = 8 \).
Since f is even, the behavior of f around x = -2 will mirror the behavior around x = 2. Thus, \( \lim_{x \to -2^-} f(x) = \lim_{x \to 2^+} f(x) = 5 \).
Conclude that \( \lim_{x \to -2^-} f(x) = 5 \) based on the symmetry of even functions.

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:
3m

Concetti chiave

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

Even Functions

An even function is defined by the property that f(−x) = f(x) for all x in its domain. This symmetry about the y-axis implies that the function takes the same value for both positive and negative inputs of the same magnitude. Understanding this property is crucial for evaluating limits and determining function behavior around specific points.
Video consigliato:
6:13
Exponential Functions

One-Sided Limits

One-sided limits refer to the behavior of a function as it approaches a specific point from one side only, either the left (−) or the right (+). In this case, lim x→2^+ f(x) and lim x→2^− f(x) indicate the values of the function as x approaches 2 from the right and left, respectively. These limits help in understanding discontinuities and the overall behavior of the function near critical points.
Video consigliato:
05:50
One-Sided Limits

Limit Evaluation at Symmetric Points

When dealing with even functions, the limits at symmetric points can be inferred from each other. For instance, if f is even and we know lim x→2^− f(x) = 8, we can deduce that lim x→−2^− f(x) will also equal 8, since f(−2) must equal f(2) due to the even property. This relationship simplifies the evaluation of limits for even functions.
Video consigliato:
05:50
One-Sided Limits
Pratica correlata
Domanda del libro di testo

Evaluate each limit. 


lim θ→0 (1/(2+sinθ)-1/2)/sin θ

388
views
Domanda del libro di testo

Determine the following limits.


Assume the function g satisfies the inequality 1≤g(x) ≤sin^2 x + 1, for all values of x near 0. Find lim x→0 g(x).

403
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→4 3(x − 4)√x + 5 / 3 − √x + 5

281
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 h→0 (5 + h)^2 − 25 / h

255
views
Domanda del libro di testo

Evaluate each limit. 


lim x→0+ 1−cos^2x / sin x

418
views
Domanda del libro di testo

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.


a. lim x→−2^+ f(x)

411
views