Overview
As we complete Chapter 2 and move on to Chapter 3, the theme of our work will focus on how we can apply the meaning of the derivative to solve important problems. Interestingly, although we use limits to define the derivative itself, it turns out that the derivative can be a useful tool in evaluating challenging limits of a certain type. In Section 2.8, we encounter a method called “L’Hôpital’s Rule” (pronounced “lo-pi-TALL”, named after French mathematician Guillaume de l’Hôpital, who published this result in 1696, but had actually acquired it from the famous mathematician Johann Bernoulli) which enables us to use a consequence of local linearity for finding difficult indeterminate limits.
Important note on the spelling and pronounciation:
In the 17th and 18th centuries, the name was commonly spelled with a silent s, “l’Hospital“, and he himself spelled his name that way. Usage of this spelling is therefore perfectly acceptable, especially in English, which usually doesn’t use the circumflex character.
However, the French spelling has changed since then: the silent ‘s’ in many words has been removed and replaced with the circumflex over the preceding vowel: “l’Hôpital“.
On the other hand, the spelling “l’Hopital” (without “s” and without
circumflex accent) should be considered incorrect, even though it
is unfortunately still seen in many places, including some textbooks!
The topics discussed in this section are: Indeterminate forms of type “\(0/0\)“. Local linearization and l’Hôpital’s Rule. Infinite limits and limits at infinity. Asymptotes. Indeterminate forms of type “\(\infty/\infty\)“.
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.
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(Review) Recall what it means to say that a limit has an indeterminate form and how such forms arise in the limit definition of the derivative.
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(Review) State the formula for the tangent line approximation to a function \(f\) at a given \(x\)-value \(a\), and write the tangent line approximation in the form
$$L_f(x) = f(a) + f'(a)(x-a).$$ -
State L’Hôpital’s Rule.
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Explain how L’Hôpital’s Rule is used to calculate limits having an indeterminate form.
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Explain what the expression \(\displaystyle \lim_{x \to \infty} f(x) = L\) means in plain English.
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State the \(\infty\) version of L’Hôpital’s Rule.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Understand the development of L’Hôpital’s Rule and how it relies upon the tangent line approximation of two functions.
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Use L’Hôpital’s Rule to calculate limits that have an indeterminate form “\(0/0\)“.
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Use L’Hôpital’s Rule to calculate limits that have an indeterminate form “\(\infty/\infty\)“.
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Use limits to identify locate horizontal and vertical asymptotes.
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Find limits involving indeterminate products (“\(0 \cdot \infty\)“): rewrite the limit in such a way that l’Hôpital’s Rule can be used.
To prepare for class
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Read the short introduction to Section 2.8 in Active Calculus prior to the Preview Activity.
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Do the Preview Activity for section 2.8 (on WeBWorK if required by your teacher).
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Read the next paragraphs in Section 2.8 in Active Calculus, stopping before Activity 2.8.2.
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Watch the following videos - and as always, make yourself a list of at least 3 things you learned from this and the following videos - and include this as part of your feedback:
After class
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Watch the following video to see how to use l’Hôpital’s Rule to deal with indeterminate limits which are products (video created at CCSL):
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Watch the following video to see how to deal with an indeterminate limit quotient involving a square root (a “radical”) - make sure to read the note shown below the video after watching it:
Important Note: In the example shown in this video, \(\frac{1}{x^2}=\sqrt{\frac{1}{x^4}}\), which works no matter if \(x\) is positive or negative. But if we had instead multiplied by \(\frac{1}{x}\) (which you need in some of your WeBWorK problems), it is important to realize that the sign of \(x\) is very important: * When \(\color{blue}{x>0}\) (e.g. when \(x\to +\infty\)), we simply have \(\color{blue}{\frac{1}{x}=\sqrt{\frac{1}{x^2}}}\), as expected. * But when \(\color{red}{x<0}\) (e.g. when \(x\to -\infty\)), we have that \(\sqrt{x^2}=|x|=-x\), and we therefore get: \(\color{red}{\frac{1}{x}=-\sqrt{\frac{1}{x^2}}}\).
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Watch the following video which explains l’Hôpital’s Rule with a graphical approach (video created at CCSL):
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Watch the following videos for more help (videos created at CCSL):
