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 2.7.48
Capitolo 2, Problema 2.7.48

Use the precise definition of infinite limits to prove the following limits.


limx→0(1x4−sin(x))=∞{\(\displaystyle\[\lim\)_{x\(\to\)0}}\(\left\)(\(\frac{1}{x^4}\)-\(\sin\]\left\)(x\(\right\))\(\right\))=\(\infty\)

Guida verificata passo dopo passo
1
Step 1: Understand the definition of an infinite limit. The limit \( \lim_{x \to a} f(x) = \infty \) means that for every positive number \( M \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( f(x) > M \).
Step 2: Consider the function \( f(x) = \frac{1}{x^4} - \sin(x) \). As \( x \to 0 \), \( \frac{1}{x^4} \to \infty \) and \( \sin(x) \to 0 \). Therefore, \( f(x) \to \infty \).
Step 3: For a given \( M > 0 \), we need to find \( \delta > 0 \) such that if \( 0 < |x| < \delta \), then \( \frac{1}{x^4} - \sin(x) > M \).
Step 4: Since \( \sin(x) \) is bounded, \( |\sin(x)| \leq |x| \). Therefore, \( \frac{1}{x^4} - \sin(x) \geq \frac{1}{x^4} - |x| \).
Step 5: Choose \( \delta \) small enough such that \( \frac{1}{x^4} - |x| > M \) for \( 0 < |x| < \delta \). This ensures that \( \frac{1}{x^4} - \sin(x) > M \), proving the limit is infinite.

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:
8m

Concetti chiave

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

Infinite Limits

Infinite limits describe the behavior of a function as the input approaches a certain value, where the function's output grows without bound. Specifically, if the limit of a function as x approaches a value results in infinity, it indicates that the function increases indefinitely in that vicinity. Understanding infinite limits is crucial for analyzing functions that exhibit vertical asymptotes or unbounded growth.
Video consigliato:
05:50
One-Sided Limits

Sine Function Behavior

The sine function, denoted as sin(x), oscillates between -1 and 1 for all real numbers x. As x approaches 0, sin(x) can be approximated by its Taylor series expansion, which shows that sin(x) behaves like x near zero. This property is essential for evaluating limits involving sin(x) and understanding how it interacts with other functions, particularly in the context of limits approaching zero.
Video consigliato:
Percorso guidato
5:53
Graph of Sine and Cosine Function

Limit Definition and Evaluation

The precise definition of a limit involves the concept of approaching a value as closely as desired. To prove limits, one often uses algebraic manipulation, substitution, or the epsilon-delta definition. In this case, evaluating the limit requires analyzing the expression as x approaches 0, particularly focusing on the dominant term in the expression, which is crucial for determining the limit's behavior.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo

Estimate the following limits using graphs or tables.


lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

422
views
Domanda del libro di testo

Evaluate each limit and justify your answer. 

lim x→4 √x^3−2x^2−8x / x−4

383
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→1 (8x+5)=13

365
views
Domanda del libro di testo

Which one of the following intervals is not symmetric about x=5?

a.(1, 9)

b.(4, 6)

c.(3, 8)

d.(4.5, 5.5)

359
views
Domanda del libro di testo

Evaluate each limit. 

lim x→2 √4x+10 / 2x−2

408
views
Domanda del libro di testo

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

279
views