MFE I - Homework 5 Due: Thursday, March 5 before 11:00pm via Gradescope Write full, clear solutions to the problems below. It is important that the logic of how you solved these problems is clear....

Due in an hour, just limits & continuity. Plz help quick


MFE I - Homework 5 Due: Thursday, March 5 before 11:00pm via Gradescope Write full, clear solutions to the problems below. It is important that the logic of how you solved these problems is clear. Although the final answer is important, being able to convey you understand the underlying concepts is more important. Please make sure the scan comes out legibly and the correct pages are associated to each question. This problem set is graded out of 28 total points. 1. Evaluate each of the following limits. (a) (2 points) lim x→3 x2 − 4 x3 − 6x2 + 9x (b) (2 points) lim x→∞ x3/2 + 1√ 7x3 + 5x 2. (3 points) Let lim x→∞ f(x) = 0 and limx→∞ g(x) = 0. For each of the following three scenarios, give possible formulas for f(x) and g(x). (a) lim x→∞ f(x) g(x) = 0 (b) lim x→∞ f(x) g(x) = 1 (c) lim x→∞ f(x) g(x) =∞ 3. (4 points) Sketch the graph of an example of a function f that satisfies all of the following conditions: • f(0) = 3 • lim x→0− f(x) = 4 • lim x→0+ f(x) = 2 • lim x→−∞ f(x) = −∞ • lim x→4− f(x) = −∞ • lim x→4+ f(x) =∞ • lim x→∞ f(x) = 3 4. (4 points) If F (x) = 5x 1 + x2 , find F ′(2) using the limit definition of a derivative. Then use it to find an equation of the tangent line to the curve y = 5x 1 + x2 at the point (2, 2). 5. (3 points) The derivative of some function g(x) is graphed below. −9 −8 −6 −4 −2 0 2 4 6 8 9 − 10 − 5 5 10 (a) On what intervals is g increasing? (b) For what x-value(s) is the tangent line for g horizontal? 6. (2 points) Suppose the tangent line to y = f(x) at the point (4, 3) passes through the point (0, 2), find f(4) and f ′(4). 7. Find lim h→0 3 √ 27 + h− 3 h in two steps: (a) (2 points) Find a function f and a number a such that f ′(a) = lim h→0 3 √ 27 + h− 3 h . (b) (2 points) Evaluate f ′(a) using the power rule. (Your answer in (b) should be a number, not an abstract expression involving a) 8. (4 points) Let f(x) =  3x 2 + 4x if x ≤ 1 2x3 + bx + c if x > 1 Find b and c so that f(x) is differentiable everywhere.