1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. x-3

Guida verificata passo dopo passo

Recall that the function \(e^x\) is an exponential function, which grows faster than any polynomial function as \(x \to \infty\).

Compare the given function \(x - 3\) to \(e^x\): since \(x - 3\) is a linear polynomial, it grows much slower than \(e^x\) as \(x\) becomes very large.

To determine growth rates, consider the limit \(\lim_{x \to \infty} \frac{f(x)}{e^x}\) for the function \(f(x) = x - 3\).

Evaluate the limit \(\lim_{x \to \infty} \frac{x - 3}{e^x}\). If this limit is 0, then \(x - 3\) grows slower than \(e^x\).

Since the limit tends to 0, conclude that \(x - 3\) grows slower than \(e^x\) as \(x \to \infty\).

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:

Concetti chiave

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

Growth Rates of Functions

Growth rates describe how functions behave as the input approaches infinity. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace. For example, polynomial functions grow slower than exponential functions like e^x as x→∞.

Video consigliato:

Percorso guidato

04:16

Intro To Related Rates

Exponential Functions

Exponential functions have the form a^x, where the variable is in the exponent. The function e^x grows rapidly and dominates polynomial and logarithmic functions as x approaches infinity. Understanding e^x is key to comparing growth rates.

Video consigliato:

6:13

Exponential Functions

Limits and Asymptotic Behavior

Limits describe the behavior of functions as the input approaches a particular value, often infinity. Asymptotic behavior focuses on how functions compare in growth by examining the limit of their ratio. This helps classify functions as growing faster, slower, or at the same rate.

Video consigliato:

Percorso guidato