This self-checking online assessment has 10 questions that will help students practice solving functions with infinite limits that converge or diverge. There are hints available on the screen, and there is an online scratchpad that students can use to work on the problems.
This informational material will explain how to evaluate integrals with infinite limits to find the value of a variable. It gives examples of integrals with limits of infinity or negative infinity that converge or diverge. There are corresponding videos available. Practice questions with a PDF answer key are provided.
This informational material will introduce students to the concept of limits, including end behavior, asymptotes, and limit notation. When learning about the end behavior of a rational function, students described the function as either having a horizontal asymptote at zero or another number or going to infinity. Limit notation is a way of describing this end behavior mathematically. There is a corresponding video available. Practice questions with a PDF answer key are provided.
This self-checking online assessment has 10 questions that will help students practice evaluating limits using graphs of functions. There are hints available on the screen, and there is an online scratchpad that students can use to work the problems.
This self-checking online assessment has 10 questions that will help students evaluate the limits of functions using algebraic substitution. There are hints available on the screen, and there is an online scratchpad that students can use to work the problems.
They're an important part of the ecosystem. They prevent disease and clean up carrion. Yet, they're also a nuisance to homeowners and a threat to livestock. Their population has recovered and grows at an incredible rate. At what point can we say that there are too many black vultures in America?
This informational material will apply a precalculus concept--limits of functions--to an environmental science issue--how biological and physical changes within an ecosystem can affect the population growth of a species. There are additional links provided for students to explore more about this issue.
In a previous read, students have learned what a limit is and how it is used to describe certain properties about functions. This interactive will challenge students to evaluate the limits of the function f(x) graphically.
In this interactive, the function f(x) will be represented by the green function.
Students will:
Finding limits for the vast majority of points for a given function is as simple as substituting the number that x approaches into the function. Since this turns evaluating limits into an algebra-level substitution, most questions involving limits focus on the cases where substituting does not work. How can you decide if substitution is an appropriate analytical tool for finding a limit?
This informational material will explain how to find a limit of a function using algebraic substitution and when this method is appropriate. There is a corresponding video available. Practice questions with a PDF answer key are provided.
This online interactive will challenge students' knowledge of limits of functions using the following scenario:
You and your friend want to understand what the definition of a limit means. Here is an informal definition of a limit.
Notation and Informal Definition of a Limit of a Function
means that as x approaches (or gets very close to) a, the function f(x) gets very close to the value L.
This video from Khan Academy on the CK-12 website will introduce students to the concept of limits by using two different functions as examples to demonstrate how to find the limits of a function.
This self-checking online assessment has 10 questions that will help students practice the skill of identifying limits of functions. There are hints available on the screen, and there is an online scratchpad that students can use to work the problems.
Humans thrive and survive within a narrow range of air pressures. When air pressures are out of this range, we have more physical problems. What happens when humans go into space? How have engineers made it possible to survive when air pressure approaches zero?
This informational material will relate the precalculus concept of limits of functions using a real-world issue--engineering astronaut spacesuits. There are embedded videos within the text.
This self-checking online assessment has 10 questions that will help students practice identifying the vertical and horizontal asymptotes of functions and identifying limits of functions. There are hints available on the screen, and there is an online scratchpad that students can use to work the problems.
Science fiction movies take it for granted that someday humans, or an alien race, will travel faster than the speed of light and build an intergalactic empire. Scientists aren't so sure that this is possible. It turns out that approaching the speed of light is very difficult. If Einstein's theories are correct, nothing that has mass can travel at the speed of light.
This informational material will apply a precalculus concept--limits of functions at infinity--to a well-known scientific theory--Einstein's theory of relativity. There is a video and links to additional information included.
