Exercise I: Continuity (2 x 15 = 30 points) For full credit, you must find and show the work for limits at every breaking point of the piecewise function. Refer to our class notes from Module 2 for similar exercises 1. At which points is the function below continuous? - 4 - 2 a 5. e -5 2. Let (a + 1)3 -x3 - 2x2 -5 T- 1 f(x) = 12rta x>1 x = 1. What values of a and b should you choose to make f continuous? Exercise II: False or false? (5 x 5 = 25 points) Consider the statement below: If lim f(x) = 6, then lim f(x) = / This is false because a function can have different limits to the left and to the right. For instance, let f(x ) = 2x (a+1 a>0. 1-0- Then lim f(x) =0, but lim f(x) = 1. The following statements are also all false. Use the same method described above to explain why by giving a counterexample using formulas or graphs. 1. If lim f(x) = 6, then f(a) = f. 2. If f is not continuous at a, then lim f(x) does not exist. 3. If lim f(x) = { and lim g(x) = 6, then f(a) = g(a). 4. If lim g(x) = 0, then lim 2 does not exist. 5. If f(x) > 0 for all r E (-0, too), then lim f(x) > 0. 2Exercise III: Limits involving infinity (2 x 10 + 5 = 25 points) 1. Compute lim a (3x - V923 + 2) ' - 2x 2. Use limits to find all the horizontal and vertical asymptotes of the function /() =2 _ 5+ 6 3. Give a formula for two functions f and g such that . f has exactly one vertical asymptote; . g has exactly two vertical asymptotes; . f + g has exactly one vertical asymptote. Make sure to justify that your choice is correct. Exercise IV: Price-to-rent ratio (4 x 5 = 20 points) The price-to-rent ratio of a house is its value divided by its annual rental price. 1. The price-to-rent ratio ratio of houses in New York City is around 37. If you rent an apartment for $2000 a month, how much do you expect it to be worth? If you buy an apartment for $500,000, how much would you rent it? 2. Is it better to rent or buy a home when the price-to-rent ratio is high? When it is low? Why? 3. Assume that you buy a home for $300,000. As the price-to-rent ratio in this area is 30, you decide to rent it for an annual rent of $10,000. Now, it is estimated that the value of your home increases by $10,000 a year, and you decide to increase the annual rent by $500 each year. Write a formula for the function p(t) describing the price-to-rent ratio of your home as a function of the number of years t. Explain. * + +00 4. Compute lim p(t) and comment, assuming that the price-to-rent ratio in the area remains 30