University of Illinois, Chicago **We aren't endorsed by this school
MATH 180
Oct 11, 2023
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Written Essay on Module 2: Limits, Continuity, and L'Hˆ opital's Rule; Due April 18 When uploading your written module essay, it is important that you have your pages for each problem in the correct order and uploaded to the correct question! Otherwise, it will receive a score of 0. You should always check your submission using a computer. Be sure that separate questions are on separate pages. Show all your work, correct answers without supporting work will not receive full credit. In the essay portions please write in full sentences! 1. (0 points) Statement of Academic Integrity This is an open note exam. You are free to use your notes to look up formulas, rules, examples, etc. Your answers must be your own, in your own handwriting and in your own words. We WILL check students' answers for solutions that have been copied in any way. You are allowed to make use of your computer/device and the internet in order to access your notes. However, you shouldn't need a calculator or the full power of the internet to answer these questions. As a result, we ask you NOT to use your computer/device/calculator/internet connection for anything other than accessing your notes. Acquiring the solution via the internet and copying it will be checked for and penalized. Note that instructors and TA's and LA's will neither be holding office hours nor answering student questions during this module's time window. While we won't hold you to the same standard as our in-person exams, we nonetheless ask that your work reflect your own understanding of the material. Again, working with others and submitting similar or identical answers WILL be checked for and will result in penalties for all parties involved. If you are found to have violated this policy, then you will receive a 0% on the entire assessment and will be reported to the Dean of Students for Academic Violations in accordance with the Student Disciplinary Policy. By signing below I acknowledge that I have read and understand the above statement and will abide by the Universitys principles regarding academic integrity. Module Essays and MyLab Module Quizzes will not be graded without your signature. Your Name: Your Signature: Page 1
Written Essay on Module 2: Limits, Continuity, and L'Hˆ opital's Rule; Due April 18 2. (10 points) Consider the following piecewise-defined function, where a is some real number: f ( x ) = xe ax e ax - 1 , x < 0 2 cos( x ) + sin( ax ) x , x > 0 . (a) (2 points) Compute lim x 0 - f ( x ) in terms of a . (b) (2 points) Compute lim x 0 + f ( x ) in terms of a . (c) (2 points) Determine the unique value of a for which lim x 0 f ( x ) exists. (d) (4 points) According to the definition of f , its domain is the set of x 6 = 0; in other words, f (0) is undefined. Explaining your thought process in full sentences, determine the unique value to which we can set f (0) that will make f continuous at x = 0. Page 2
Written Essay on Module 2: Limits, Continuity, and L'Hˆ opital's Rule; Due April 18 3. (6 points) (a) (2 points) Compute the following limit: lim x →∞ x 2 · ln cos 1 x (b) (4 points) In a few complete sentences, explain how you did the computation. Describe what the limit looked like at first, any manipulation you did, and the result (as well as any intermediate steps). Page 3
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