Overview
In this section we cover types of limits that we have not seen before, including limits containing an absolute value, indeterminate differences of the type \(\infty - \infty\), and indeterminate products and powers. To deal with these, we will study different techniques such as the use of the conjugate (a technique that we have already seen in Chapter 1), moving the limit inside a continuous function, and the Squeeze Theorem.
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.
- Express the absolute value of a function as a piecewise function and and use this to evaluate limits that involve the absolute value of an expression.
- Use the conjugate to evaluate indeterminate differences of the type \(\infty - \infty\).
- State the Squeeze Theorem and explain why it is true using a graph.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Evaluate the limit \(\lim_{x \to a} f(g(x))\) of a composite function by moving the limit inside the outer function, that is \(\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x))\), in the case where \(b = \lim_{x \to a} g(x)\), and \(f\) is continuous at \(x=b\).
- Find the limit of an indeterminate power by using \(A^B=e^{\ln(A^B)}=e^{B\ln(A)}\) and applying l’Hôpital’s Rule to the exponent (rewritten as a fraction).
- Realize which forms of limits are not indeterminate, i.e. which always result in a clear answer (e.g. “\(0^\infty\)“\(=0\), “\(\ast / \infty\)“\(=0\), “\(\ast / 0^+\)“\(=\infty\))
- Find the limit of a function using the Squeeze Theorem.
- Use l’Hôpital’s Rule and the Squeeze theorem to rank functions based on how fast they approach infinity as \(x\) approaches infinity.
To prepare for class
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Read the beginning of Section 2.9: This is an additional section written by C. Fortin and G. Indurskis which does not exist in the “official” version of Active Calculus. You can download an electronic copy in pdf-format here.
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Do the first two problems of the Preview Activity on WeBWorK, referring to the detailed solutions in Example 2.5 in section 2.9 as needed.
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Watch the third problem in this video at 5:28 where the conjugate is used to evaluate a limit of the form \(\infty - \infty\). (You may review the beginning of this video as necessary. This video was created at CCSL):
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Do the third problem of the Preview Activity on WeBWorK.
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Watch the following videos to see when we can move the limit of a composite function inside the outer function (videos created at CCSL):
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Do the last two problems of the Preview Activity on WeBWorK using a graph of the outside function to help you find the correct answers.
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Watch the following video to see how to apply this technique to determine an indeterminate limit which is a power (note that in the video, the form “\(0^\infty\)” is incorrectly listed as indeterminate, even though it is in fact simply \(0\)):
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Watch the following video on the Squeeze Theorem:
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Use the Squeeze Theorem to find \(\displaystyle \lim_{x \to \infty} \frac{\sin x}{x}\). (This is in fact done in detail in section 2.9, at the beginning of the subsection Using the Squeeze Theorem to Evaluate Limits, but you should try to attempt this on your own before looking at the solution.) As part of your pre-class feedback, write a short sentence explaining why you cannot use l’Hôpital’s Rule to find this limit.
