Please answer the following:

\f(1 point) Suppose that lim f(x) = 0, lim g(x) = 0, lim h(x) = 1, limp(a) = co, and lim q(a) = 00. Evaluate each of the following limits. (a) lim If (z) ]9(z) = (b) lim If (a) P(z) _ (c) lim [h(z)]P(z) = (d) lim [p(z) ] f(z) = (e) lim [p(a) ]9(z) = (f) lim () p(x) = Note: Input DNE, infinity, and -infinity for does not exist, oo, and -oo, respectively. If the result is indeterminate, enter I.. . - _ ('1 point} Consider the function x) 42: + 1' (a) Find all critical numbers :2 of f. c = (b) f is concave up for a: E (c) f is concave clown for a: E (d) Using the 2nd derivative test, the local maxima of f occur at a: = (e) Using the 2nd derivative test, the local minima of 3' occur at a: = Note: Input U, innity, and -innity for union, 00, and oo, respectively. If there are multiple answers, separate them by commas. If there is no answer, input none. (1 point) Suppose g(a) = (f(a) ) where f is positive and concave up for all r E 1. (a) f" (a) > on I. (b) g"(a) = 2(A2 + B f" (x)), where A = and B = (c) g"(x) > on I. (d) g(a) is on I. Note: Input CU, CD, f(x), f(x), and f"(x) for concave up, concave down, f(a), f' (a), and f" (a), respectively.\f(1 point) For what values of a and b does lim sin(2x) b tat = 0? .2 a = b =(1 point) Consider the function f (a) = a2 - 4a + 6 on the interval [0, 4]. (a) What conditions must hold true in order to apply Rolle's Theorem? f(a) is ? v on [0, 4]; changing, continuous, positive, negative, differentiable f (x) is ? von (0, 4) ; and f (0) = f(4) = (b) Since part (a) holds true for f (), by Rolle's theorem, there exists a c such that: f' ( c ) = (c) Find c. C =(1 point} Use Rolle's Theorem and a proof by oontradiction to show that the function at) = 8:c7 82: 12 does not have two real roots. Proof: Suppose x) has two real roots at and b such that u) = b) 2 Since the conditions of Rolle's theorem hold true for f on [11, b], there exists at least one number c in the interval (a, 1;) such that f" [c] = changinglnegativefzerofpositivefundened he However, the derivative f'(:c) = is always ? v and, therefore, it is _. ?_ v for {'(m) = _ _ . [inllkelyfposmbleflmpDSSIbleianSIble 0951 does "Qt This oontraclicts the conclusion of Rolle's Theorem and, therefore, f ? v have two real roots. ('1 point} Consider the function x) = 22:3 23 on the interval [3, 3]. (a) The slope of the secant linejoining (3, f{3)) and (3, f(3)) is m = (b) Since the conditions of the Mean Value Theorem hold true, there exists at least one c on (3, 3) such that f' (c) = (c) Find c. Note: If there is more than one answer, separate them with a comma. (1 point} Find all critical numbers 4:: of t) = 2933 + W3. Note: If there is more than one critical number, separate them by a comma.