\fProblem 6. Evaluate the limit lim (-5+ h)2- 25 Challenge Problems: Note that the final page of the recitation activity will usually have problems such as these, which go beyond h -0 h the sorts of problems that you are expected to know on quizzes or exams. Depending on time, your TA may choose to cover these problems, or might opt to skip them. Some students may find them interesting to do on their own, but they are not required. These final two questions are related. They involve some review of trigonometry and the second one is more challenging. Struggling with these questions is a great way to deepen your understanding of the course material and gain mastery of these topics Problem 8. This problem has an associated Desmos graph here https://www.desmos.com/calculator/snqy6xi6hs. You can use the slider to choose different values for b and the calculator will automatically compute the value of sin (" ) at some of the values of 0 listed below. 1. Estimate lime-o sin (" ) by plugging in 0 = .1, .01, .001, -.1, -.01, -.001 2. Now estimate lime-,o sin ( ) by plugging in 0 = 3, 3, 13. 17, -3, -7, -13' -17 3. What do you notice about your answers from the previous two parts? After doing both of them do you know the value of lime-o sin ( )? What limitations of computing limits via plugging in values does this reveal? Problem 9. Billy is trying to evaluate the following limit 4 Problem 7. Evaluate the limit lim 7th 7 lim(x2 + 4) sin (). h - C h x-0 He explains "The limit is equal to 4 by the squeeze theorem. First we know that lim (x2 + 4) = lim x2 + lim 4 (sum limit rule) x-0 lim x2 + lim 4 = (lim x) + lim 4 power limit rule) lim x + lim 4 = 0 + lim 4 (basic limit rule) x-+0 x-+0 0 + lim 4 = 4 (basic limit rule). X-+0 So lim-0(x2 + 4) = 4. We know that sin(0) is always trapped between -1 and 1, so sin (" )is too. Therefore by the squeeze theorem lim(x2 + 4) sin ( ) =4." Is Billy right? If not what mistakes did he make