2. Assume we know the following about the function h: |x 1| < 0.2 =) |h(x) (a) Can you guarantee that there exists |x > 0 such that =) |h(x) 1| < 4| < 1. 4| < 0.5? Answer yes or no. If the answer is yes, also provide one such . If the answer is no, justify with an example (a sketch of a graph or an equation is enough). (b) Can you guarantee that there exists |x > 0 such that =) |h(x) 1| < 4| < 1.5? Answer yes or no. If the answer is yes, also provide one such . If the answer is no, justify with an example (a sketch of a graph or an equation is enough). (c) Let's assume that we also know that the limit L = lim h(x) exists. Based on x!1 what you know so far, what can we say about the value of L? Why? 3. Prove that lim x!1 x2 1 1 = . +2 3 Write a proof directly from the "- definition of limit. Do not use any of the limit laws, or any other results. For examples of these proofs, you may want to re-watch videos 2.6 and 2.7, or you may want to re-read Examples 1, 2, 6, and 7 in section 2.2 of the textbook. 4. Let f be a function with domain R. Let a, L 2 R. Assume that lim f (x) = L. x!a Prove that lim f (2x) = L. x!a/2 Write a formal proof directly from the "- definition of limit. Do not use any of the limit laws, or any other results. For examples of these proofs, you may want to re-watch videos 2.10 and 2.12, or you may want to re-read the proof of Theorem 2.3.2 in the book