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}…

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.…

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.…

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