Tag Archives: M32A

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A tricky limit from the Math 32A final


The following limit appeared on yesterday’s Math 32A final:
\[
\lim_{(x, y) \to (0, 0)} \frac{\sin x^2 + \tan^2 y}{x^2 + \tan y^2}.
\]
The limit seemed to cause a lot of trouble. In fact, no one in the lecture of 200 students had a perfect solution! I thought I would write up a solution here to give some closure on the problem. I will use only tools that appear in a standard calculus course: the squeeze theorem and the fact that \(\lim_{x \to 0} \sin(x) / x = 1\).

The first thing to note is that evaluating the limit along the \(x\)-axis (taking \(y = 0\)) gives a limit of \(1\):
\[
\lim_{x \to 0} \frac{\sin x^2}{x^2} = \lim_{u \to 0} \frac{\sin u}{u} = 1.
\]
This tells us that if the original limit exists, it must be \(1\). However, this argument does not allow us to conclude that the limit does indeed exist. Choosing any other path through \((0, 0)\) will also give a limit of \(1\), so one should anticipate that the limit exists and is equal to \(1\). We remark that, in general, \(\lim_{(x, y) \to (a, b)} f(x, y) = L\) if and only if \(\lim_{(x, y) \to (a, b)} |f(x, y) – L| = 0\). This reformulation is convenient because \(0 \leq |f(x, y) – L|\), so applying the squeeze theorem is less fussy: we only need to bound \(|f(x, y) – L| \leq g(x, y)\) where \(\lim_{(x, y) \to (a, b)} g(x, y) = 0\). Thus, we wish to show that
\[
\lim_{(x, y) \to (0, 0)} \left| \frac{\sin x^2 + \tan^2 y}{x^2 + \tan y^2} – 1\right| = 0.
\]
To this end, we compute the following bound
\[
\begin{align*}
\left| \frac{\sin x^2 + \tan^2 y}{x^2 + \tan y^2} – 1\right| &= \left| \frac{\sin x^2 + \tan^2 y}{x^2 + \tan y^2} – \frac{x^2 + \tan y^2}{x^2 + \tan y^2}\right|\\
&= \left| \frac{\sin x^2 – x^2 + \tan^2 y – \tan y^2}{x^2 + \tan y^2}\right|\\
&\leq \left| \frac{\sin x^2 – x^2}{x^2 + \tan y^2}\right| + \left| \frac{\tan^2 y – \tan y^2}{x^2 + \tan y^2}\right|\\
&\leq \left| \frac{\sin x^2 – x^2}{x^2}\right| + \left| \frac{\tan^2 y – \tan y^2}{\tan y^2}\right|.
\end{align*}
\]
The first inequality holds by the triangle inequality (\(|a + b| \leq |a| + |b|\)), while the second inequality holds because \(\tan y^2 \geq 0\) (when \(y\) is close to \(0\)) and \(x^2 \geq 0\). Since we have
\[
0 \leq \left| \frac{\sin x^2 + \tan^2 y}{x^2 + \tan y^2} – 1\right| \leq \left| \frac{\sin x^2 – x^2}{x^2}\right| + \left| \frac{\tan^2 y – \tan y^2}{\tan y^2}\right|,
\]
it suffices to show that the limit of the expression on the right is \(0\) as \((x, y) \to (0, 0)\). The expression on the right is much easier to deal with than the original limit because it is the sum of two terms, each of which only involves a single variable. Thus, one can use familiar methods from single variable calculus to compute the limits. One can show that in fact we have
\[
\lim_{x \to 0} \left| \frac{\sin x^2 – x^2}{x^2}\right| = 0 \quad\text{and}\quad \lim_{y \to 0} \left| \frac{\tan^2 y – \tan y^2}{\tan y^2}\right| = 0.
\]
The first limit follows easily from the fact that \(\lim_{x \to 0} \sin(x) / x = 1\), while the second limit requires a little more ingenuity. We compute
\[
\begin{align*}
\lim_{y \to 0} \left| \frac{\tan^2 y – \tan y^2}{\tan y^2}\right| &= \left| \lim_{y \to 0} \frac{\tan^2 y}{\tan y^2} – \lim_{y \to 0} \frac{\tan y^2}{\tan y^2} \right|\\
&= \left| \lim_{y \to 0} \frac{\tan^2 y}{\tan y^2} – 1 \right|\\
&= \left| \lim_{y \to 0} \frac{\tan^2 y}{y^2} \frac{y^2}{\tan y^2} – 1 \right|\\
&= \left| \lim_{y \to 0} \left(\frac{\tan y}{y}\right)^2 \frac{y^2}{\tan y^2} – 1 \right|\\
&= | (1)^2 (1) – 1 |\\
&= 0.
\end{align*}
\]
The penultimate equality follows from the fact that \(\lim_{y \to 0} \tan(y) / y = 1\), which can be deduced by writing \(\tan y = (\sin y) / (\cos y)\) and using the familiar \(\sin(y) / y\) limit identity (or an application of L’Hopital’s rule). At any rate, we may now deduce from the squeeze theorem that
\[
\lim_{(x, y) \to (0, 0)} \frac{\sin x^2 + \tan^2 y}{x^2 + \tan y^2} = 1.
\]
Whew.

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Math 32A Final Review Problems

I have just finished putting together some problems for the final review for Math 32A, which are available here. I have included answers to the problems. I haven’t gotten a chance to double check my work on the solutions, so let me know in the comments below if you disagree with any of the answers. I will go over the solutions this evening again and update the answers if I find any problems.

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Math 32A Midterm 2 Review Questions

I have compiled a few review questions for the second midterm for Math 32A, which are available here. The questions are on the harder side of what I would expect to see on the actual exam, but hopefully you find them interesting and/or helpful in your studying. Let me know in the comments below if you have questions about any of the problems.

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Math 32A/H Final Review Materials


I have started compiling some review materials for math 32A and 32AH for this quarter. They consist of

I will post solutions to the exercises before the review session on Thursday. As always, let me know in the comments below if anything is unclear or incorrect.

Update I have posted the solutions to the practice problems. Also, a student pointed out a typo in statement of the final problem in the review. Spherical coordinates should be given by
$$
x = \rho \cos \theta \sin \varphi, \quad y = \rho \sin \theta \sin \varphi, \quad z = \rho \cos \varphi.
$$
These changes have been made in the version of the problems and solutions online.

Update 2 There was a mistake in the solution of problem 10 involving curvature. This has (hopefully) been fixed.

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32A Midterm 2 Review Materials

I have written up some review materials for the second midterm for Math 32A. They include:

  1. Overview
  2. Review questions
  3. Review solutions

I think the questions and solutions cover the majority of the conceptual material likely to appear on the midterm. I’ve taken some of the problems/solutions from previous times I’ve taught this course, so some of the notation may not agree with what you are used to — so be warned.

As usual, please let me know in the comments below if anything is unclear or incorrect.

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Math 32A Review Materials

I have finished compiling some review materials for Math 32A this quarter. The materials include:

As usual, let me know in the comments below if you notice any errors or if anything is unclear.

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Math 32A Midterm II Review

I’ve written up some review questions for the second midterm for Math 32A this quarter. Here is a link to the questions. Here are the solutions. As usual, let me know in the comments below if you find any mistakes or if anything is unclear.

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Math 32A Midterm 1 Review Questions


I have compiled some review materials for the first midterm for Math 32A this quarter. Here are links to the questions and solutions. As always, feel free to let me know in the comments below if you have any questions or notice any errors.

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Math 32A Final Review

I just uploaded a review for the final exam for Math 32A (multivariable differential calculus) which is available here. If you find typos or think that some portion needs clarification, please let me know in the comments below. The review sheet contains an overview of the topics covered in the course, as well as many examples worked out in full detail.

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Math 32A Midterm 1 Review

I wrote up a couple of review questions for the first midterm for Math 32A this quarter. The questions without solutions are available here, and full solutions here. There are two multipart questions, which cover the majority of the material we’ve covered in the course so far.