Overview
We are returning to study limits of general functions, and examine how we can evaluate them. In particular, we will see that functions which are “nice” in a certain way (we will call such functions “continuous”) allow us to evaluate limits very easily. Knowing which functions are continuous or where they fail to be continuous will then enable us to focus on the actually “interesting” parts of functions, where we actually need to consider a limit.
This section covers the following concepts: Limit Laws and how to use them to evaluate limits algebraically, One-sided limits, Continuity, Direct Substitution Property, Evaluating (one-sided) limits algebraically.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Explain what is a left-hand limit and right-hand limit.
- Explain the difference between a limit and a one-sided limit.
- Estimate the limit from the left and the limit from the right of a function at a point on a graph (or determine that it does not exist).
- List the six fundamental properties of limits, the so-called “Limit Laws”.
- State the definition of a continuous function \(f\) at a point \(x=a\) and on an interval \([a,b]\).
- Determine whether a function is continuous at a point by examining the graph of that function.
- List different types of functions (polynomials, rational, etc) that are known to be continuous on their domain.
- Evaluate where a function is continuous by finding its domain.
- If \(f\) and \(g\) are continuous functions on their domain, list functions related to \(f\) and \(g\) (for example \(f+g\)) that are also continuous on their domain.
- Evaluate the limit of a function which is known to be continuous at a given point using direct substitution.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Explain how the domain of a rational function is related to finding the limit of a rational function.
- Determine the limit of a function at a point using the algebraic formula (or determine that it does not exist). This includes using the different rules of a piecewise-defined function to determine the one-sided limits at the transition points.
- Provide examples (both graphically and numerically) of functions where both one-sided limits exist at a point, and yet the limit at that point fails to exist.
- Verify algebraically whether a function is continuous at a specific point.
- Given an algebraic expression for a function \(f\), find where it is continuous.
- Explain what makes a discontinuity removable - graphically and in your own words.
- Given a function that has a removable discontinuity, redefine this function to make it continuous.
- Provide examples of real-life functions that are not continuous.
To prepare for class
- Watch the 2 videos about the so-called Limit Laws (“Lesson Part 1 and 2”) on this page by the University of Waterloo, and (while watching the first video) try to sketch a picture which illustrates each of the Limit Laws which are presented to you.
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Do the 4 quiz questions on the same page (this will count as your Preview Activity for this class) - it will immediately tell you if you have the correct answers (or how to correct them) on the same page. Optionally, you may also choose to do the exercises (which have answers and also detailed solutions on the same page).
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Watch the following video which explains how to find (one-sided) limits of a piecewise-defined function both graphically and algebraically (video created at CCSL):
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Watch the 3 videos (“Lesson Part 1-3”) about Continuous Functions on this page by the University of Waterloo.
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Read subsection 1.7.2 in Active Calculus: “Being continuous at a point” and read carefully Definition 1.7.6 which defines continuity. Make sure to understand how this can be informally understood as saying that the graph “has no break at a given point” - and how this in turn can be used to evaluate a limit, if one already knows that the function is continuous: This is often called the “Direct Substitution Property” of continuous functions.
After class
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Watch the following video which studies continuity both using a graph and the algebraic expression of a function:
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Watch the following video which gives a summary of properties of continuous functions (video created at CCSL):
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Watch the following video which shows how to determine if a piecewise-defined function is continuous (video created at CCSL):
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(Re-)Read the first half of section 1.7, up to (but not including) subsection 1.7.3 (“Being differentiable at a point”).
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Do some experimentation with the following interactive applets:
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Finish any in-class activities you might not have finished during class.
- Do the problems on the WeBWorK assignment for this section.
