Which functions are of exponential orders? part2

TL;DR

Functions with exponential behavior can be determined by checking the limits as T goes to infinity and manipulating the exponents.

Transcript

okay it's two more examples we are gonna see if these functions are of exponential waters were not let's look at that first one and just like in the previous video we are going to check the limit right so we put on check and we take the limit as T goes to infinity and then we put this on the top e to the T squared over T plus 1 and we put down e to... Read More

Key Insights

• 🫤 Exponential behavior in a function can be determined by checking the limits as T goes to infinity and manipulating the exponents.
• 😑 Long division can be used to simplify the expression and determine the correct exponent of a function.
• ⛔ The choice of alpha value can influence whether the limit of a function is equal to zero and exhibits exponential behavior.
• 👻 The sine function's range of -1 to 1 allows for more flexibility in selecting a suitable value for alpha and confirming exponential behavior.
• ♾️ Understanding the properties of exponential functions and their behavior at infinity is essential in analyzing their exponential behavior.
• 🧡 By considering the range and behavior of the functions, we can confidently determine if they exhibit exponential behavior or not.
• ⛔ Exponential behavior in functions can be confirmed by selecting appropriate values for alpha that make the limit of the function equal to zero.

Questions & Answers

Q: How can we determine if a function exhibits exponential behavior?

To determine exponential behavior, we check the limits as T goes to infinity and manipulate the exponents of the function to simplify the expression.

Q: How is long division used to analyze a function for exponential behavior?

Long division is used to simplify the expression by dividing the exponent of the function by the denominator. This allows us to determine the correct exponent and confirm exponential behavior.

Q: How do we select a value for alpha in the second function analysis?

In the second function analysis, we select a value for alpha that makes the limit of the function equal to zero. By choosing the correct value, we can confirm exponential behavior.

Q: How does the analysis of the third function differ from the other two?

In the analysis of the third function, we consider that the sine function is at most equal to one. Therefore, by selecting a suitable value for alpha, we can ensure that the limit of the function is equal to zero and exhibits exponential behavior.

Summary & Key Takeaways

• The analysis examines three different functions to determine if they exhibit exponential behavior.

• The first function is evaluated using long division to simplify the expression and determine the exponent.

• The second function is analyzed by selecting a value for alpha that makes the limit equal to zero.

• The third function is evaluated based on the fact that e to the at most one is just at most e.