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 2.59
Capitolo 2, Problema 2.59

Evaluate limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\)\(\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}.


f(x)=6ex+203ex+4f\(\left\)(x\(\right\))=\(\frac{6e^{x}\)+20}{3e^{x}+4}

Guida verificata passo dopo passo
1
Identify the function for which you need to find the limits: \( f(x) = \frac{6e^{x} + 20}{3e^{x} + 4} \).
To evaluate \( \lim_{x \to \infty} f(x) \), divide the numerator and the denominator by \( e^x \), the highest power of \( e \) in the expression. This simplifies the function to \( \frac{6 + \frac{20}{e^x}}{3 + \frac{4}{e^x}} \).
As \( x \to \infty \), the terms \( \frac{20}{e^x} \) and \( \frac{4}{e^x} \) approach 0 because \( e^x \) grows exponentially. Thus, the expression simplifies to \( \frac{6}{3} \).
Now, evaluate \( \lim_{x \to -\infty} f(x) \). In this case, \( e^x \to 0 \) as \( x \to -\infty \), so the function simplifies to \( \frac{20}{4} \).
Conclude by stating the limits: \( \lim_{x \to \infty} f(x) = 2 \) and \( \lim_{x \to -\infty} f(x) = 5 \).

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.

Limits at Infinity

Limits at infinity describe the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine horizontal asymptotes and the end behavior of functions.
Video consigliato:
03:07
Cases Where Limits Do Not Exist

Exponential Functions

Exponential functions, such as f(x) = (6e^x + 20)/(3e^x + 4), are characterized by a constant base raised to a variable exponent. These functions grow rapidly as x increases, making them essential in calculus for evaluating limits and understanding growth rates.
Video consigliato:
6:13
Exponential Functions

Rational Functions

Rational functions are ratios of polynomials, which can exhibit unique behaviors at different values of x. In the context of limits, analyzing the degrees of the numerator and denominator helps determine the limit as x approaches infinity or negative infinity, revealing important characteristics of the function.
Video consigliato:
6:04
Intro to Rational Functions
Pratica correlata
Domanda del libro di testo

Determine the following limits.


limθ→0−sinθcos2θ−1{\(\displaystyle\[\lim\)_{\(\theta\]\to\)0^{-}}\(\frac{\sin\theta}{\cos^2\theta-1}\)}

297
views
Domanda del libro di testo

Determine the following limits.

lim x→0^− 2 / tan x

492
views
Domanda del libro di testo

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


limx→0(1x2+1)=∞{\(\displaystyle\)\(\lim\)_{x\(\to\)0}}\(\left\)(\(\frac{1}{x^2}\)+1\(\right\))=\(\infty\)

409
views
Domanda del libro di testo

Complete the following steps for each function.


c. State the interval(s) of continuity.


f(x)={2x if x<1

x^2+3x if x≥1; a=1

347
views
Domanda del libro di testo

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

f(x)=−3e−xf\(\left\)(x\(\right\))=-3e^{-x}

358
views
Domanda del libro di testo

Determine the following limits.


limx→1+x2−5x+6x−1{\(\displaystyle\)\(\lim\)_{x\(\to\)1^{+}}}\(\frac{x^2-5x+6}{x-1}\)

322
views