Domanda del libro di testo
Even function limits Suppose f is an even function where lim x→1^− f(x)=5 and lim x→1^+ f(x)=6. Find lim x→−1^− f(x) and limx→−1^+ f(x).
374
views
Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 96
Find the horizontal asymptotes of each function using limits at infinity.
f(x) = (3e5x + 7e6x) / (9e5x + 14e6x)
Guida verificata passo dopo passo1
Identify the highest power of the exponential function in both the numerator and the denominator. In this case, it's \( e^{6x} \).
Factor \( e^{6x} \) out of both the numerator and the denominator.
Rewrite the function as \( f(x) = \frac{e^{6x}(\frac{3}{e^x} + 7)}{e^{6x}(\frac{9}{e^x} + 14)} \).
Cancel \( e^{6x} \) from the numerator and the denominator, simplifying the expression to \( f(x) = \frac{\frac{3}{e^x} + 7}{\frac{9}{e^x} + 14} \).
Evaluate the limit of \( f(x) \) as \( x \to \infty \). As \( x \to \infty \), \( \frac{3}{e^x} \to 0 \) and \( \frac{9}{e^x} \to 0 \), so the limit becomes \( \frac{0 + 7}{0 + 14} = \frac{7}{14} = \frac{1}{2} \). Thus, the horizontal asymptote is \( y = \frac{1}{2} \).
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:
7mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a function as the input approaches infinity or negative infinity. They indicate the value that the function approaches, providing insight into its long-term behavior. To find horizontal asymptotes, one typically evaluates the limit of the function as x approaches infinity.
Video consigliato:
5:46
Graphs of Exponential Functions
Limits at Infinity
Limits at infinity involve determining the value that a function approaches as the variable grows indefinitely large or small. This concept is crucial for analyzing the end behavior of functions, especially rational functions, where the degrees of the numerator and denominator can dictate the limit's outcome.
Video consigliato:
05:50One-Sided Limits
Exponential Functions
Exponential functions, such as e^(kx), grow or decay at rates proportional to their current value. In the context of limits, the behavior of these functions as x approaches infinity is significant, as they can dominate polynomial terms, influencing the overall limit and thus the horizontal asymptote of the function.
Video consigliato:
6:13Exponential Functions
Pratica correlata
Domanda del libro di testo
Let g(x)= {1 if x≥0
−1 if x<0.
a. Write a formula for |g(x)|.
366
views
Domanda del libro di testo
Sketch a graph of y=2^x and carefully draw three secant lines connecting the points P(0, 1) and Q(x,2^x), for x=−3,−2, and −1.
430
views
Domanda del libro di testo
Find functions f and g such that lim x→1 f(x)=0 and lim x→1 (f(x)g(x))=5.
421
views
Domanda del libro di testo
Find the horizontal asymptotes of each function using limits at infinity.
f(x) = (2ex + 3) / (ex + 1)
373
views
Domanda del libro di testo
Evaluate lim x→1 3√x − 1 / x (Hint: x−1=(3√x)^3−1^3.)
366
views