1.2: Limits of Functions (including one-sided limits)
There are limits to many things around us, including algebraic functions. Limits can help us find values that are not possible in other aspects of mathematics, including the velocity of a falling object as well as the area under curves. Other examples include the population of certain animals, depending on predators and food sources, as well as the instant speed of an object. Limits allow us to explore what occurs with a function when the dependent variable approaches a certain value.
Subunit 1.2 discusses the fundamental concept of a limit, which is a critical definition for calculus. This subunit will require you to find limits, including one-sided and infinite limits, graphically and algebraically. This subunit also helps you identify the properties of limits and cases where limits do not exist.
1.2.1: Finding Limits
Read this section on approximations to start understanding the concepts of limits. This reading discusses the tangent line to a graph and approximate area under a curve. Then, complete review questions 1-8 toward the bottom of the page. These exercises will provide you with the opportunity to apply concepts of a point tangent to a graph and approximate the area under curves. The solutions to these problems are located here.
Take notes as you watch these videos. Listen to the presentation carefully until you are able to understand the concept of a limit in calculus and the formal definition of a limit.
Read this section on finding limits. This reading discusses how to find a limit numerically, using a graph, identify nonexisting limits, and use the formal definition of a limit. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to find limits algebraically, graphically, and using tables. The solutions to these problems are located here.
1.2.2: Evaluating Limits
Take notes as you watch this video. Listen to the presentation carefully until you are able to recall and apply limit properties.
Read this section on finding limits. This reading discusses how to find a limit of various functions, the properties of limits, and using the squeeze theorem to find special limits.
Complete exercises 1-2 and 7-26. These exercises will provide you with the opportunity to find limits algebraically, graphically, and using tables. The solution for each problem can be found by clicking on the gray triangle beside each problem.
1.2.3: Infinite Limits
Take notes as you watch these videos. Listen to the presentation carefully until you are able to recall and apply properties of limits dealing with infinity and to find limits of positive and negative infinity.
Read this section on finding limits. Watch all of the videos in the section as well. This reading discusses how to find infinite limits of functions and the properties of infinite limits, identify asymptotes of functions, and analyze end behavior of functions. Then, complete review questions 8-10 toward the bottom of the page. These exercises will provide you with the opportunity to analyze the end behavior of the graph and find the asymptotes. The solutions to these problems are located here.
Complete exercises 27-36 toward the bottom of the page. These exercises will provide you with the opportunity to find limits algebraically, graphically, and using tables. The solution for each problem can be found by clicking on the gray triangle beside each problem.