(5) Our goal in this exercise is a first attempt at making precise our intuition surrounding limits at infinity. (a) (1 mark) Give an informal one sentence explanation of your understanding of the statement The limit of 2x + 1 as a goes to oo is co. (b) (3 marks) For x, r E R, let p(x, r) be the condition 2x + 1 > r Find a value of k so that p(x, 10) is true for every > > k. Justify your response. (c) (1 mark) Find a value of k so that p(x, 31) is true for every r > k. Justify your response. (d) (1 mark) Find a value of k so that p(x, 71) is true for every > > k. Justify your response. (e) (0 marks ) Find a value of k so that p(x, 10001) is true for every > > k. Justify your response. (f) (2 marks) Find a formula for k as a function of r so that p(x, r) is true for every x > k. Justify your response. (g) (1 mark) Informally, your formula in the previous part allows to verify the following fact: for any r E R, f(x) = 2x+1 eventually gets bigger and stays bigger than r. Consider the following definition: Definition. Let f : R - R be a function. We say the limit of f(x) as x goes to co is co when for every r E R there exists k E R so that f(x) > r for every x > k. Are you convinced the limit of f(x) = 2x + 1 as a goes to co is co, as defined by the definition above? Explain your thinking with a sentence or two by referring to your answer from (f) and the definition in (g). (It is okay if the answer is "No, I am very confused by this question".) (h) (3 marks) Given the definition of the limit of f(x) as a goes to co is co given above, what do you expect the definition of the limit of f(x) as a goes to co is -co to be? Give your answer both as a sentence and as a statement using the language of formal logic