1. Calculate these limits, explaining your answer. (6) a) lim e*(x + sin x) x-+ 00 x2 + x3ex x*+ x- In x b) lim x-+0o x3 - 7x2 + 9x + 45 2. Calculate these derivatives, explaining your steps. (8) a) a x sin x + (x - 3)4 In x dx 2 b) x sin x dx2 3. Calculate these integrals, explaining your steps. (8) a ) 1 8 + -dx V1 - x2 X V2IT b) 4x sin (x+ 7 dx 4. Integrals are one of the three main pilars of calculus. They calculate areas under curves. Explain why algebra can't solve this problem. What is the barrier to this calculation? Why are limits required? (4)