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 2.21
Capitolo 2, Problema 2.21

Determine the following limits. 
lim x→∞ (3x12 − 9x7)

Guida verificata passo dopo passo
1
Identify the highest power of x in the expression, which is x^{12}.
Factor out x^{12} from the expression: 3x^{12} - 9x^{7} = x^{12}(3 - 9x^{-5}).
Rewrite the limit expression: \( \lim_{x \to \infty} x^{12}(3 - 9x^{-5}) \).
Evaluate the limit of the factored expression: \( \lim_{x \to \infty} x^{12} \) and \( \lim_{x \to \infty} (3 - 9x^{-5}) \).
Since \( x^{12} \to \infty \) and \( 3 - 9x^{-5} \to 3 \), the overall limit is determined by the behavior of x^{12}.

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:
2m

Concetti chiave

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

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave for very large values of x, particularly in polynomial functions where the highest degree term often dominates the behavior of the function.
Video consigliato:
05:50
One-Sided Limits

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the context of limits, the degree of the polynomial plays a significant role in determining the limit as x approaches infinity, as the term with the highest degree will have the greatest impact on the function's value.
Video consigliato:
6:04
Introduction to Polynomial Functions

Dominant Term

The dominant term in a polynomial is the term with the highest degree, which significantly influences the value of the polynomial as x becomes very large. When calculating limits at infinity, identifying the dominant term allows for simplification, as lower degree terms become negligible compared to the dominant term.
Video consigliato:
2:02
Simplifying Trig Expressions Example 1
Pratica correlata
Domanda del libro di testo

Consider the position function s(t)=−16t^2+100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=3. <IMAGE>

430
views
Domanda del libro di testo

Determine the following limits.

lim x→∞ (5 + (cos4 x) / (x2 + x + 1))

356
views
Domanda del libro di testo

Determine the following limits.


limx→0−csc(x){\(\displaystyle\[\lim\)_{x\(\to\)0^{-}}}\(\csc\]\left\)(x\(\right\))

354
views
Domanda del libro di testo

a. Analyze limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\)\(\lim\)_{x\(\to\)-\(\infty\)}{f(x)}} for each function.


f(x)=1+x−2x2−x3x2+1f\(\left\)(x\(\right\))=\(\frac{1+x-2x^2-x^3}{x^2+1}\)

343
views
Domanda del libro di testo

Determine the interval(s) on which the following functions are continuous. 

p(x)=3x^2−6x+7 / x^2+x+1

286
views
Domanda del libro di testo

Determine the following limits at infinity.


lim t→∞ (−12t^−5)

422
views