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MA103 – Midterm Test2013 FallPage 1 of 21. Without using L’Hˆopital’s Rule, evaluate the following limits if they exist:[10marks](a) limx→2x2−x−2x−2(b)limx→5-2x−10|x−5|(c) limx→0√x+25−5x(d)limx→∞√4x4+3x2−3x3/2+22.f(x) is defined asf(x) =braceleftbiggeAxwhenx<0,2x+Bwhenx≥0.For what values ofAandB[5 marks]willf(x) be continuous and differentiable everywhere?3.(a) State the limit definition of the derivative.[3 marks](b) Using the limit definition of the derivative, compute the derivative for the function[5 marks]f(x) =x2+ 2x-1.4. For each of the following, findy′=dydxusing any method you wish. Leave your answer[12marks]in terms ofxorxandy. Do NOT spend time simplifying your answers.(a)y=x23x(b)y=2tan(x)1+x2(c)y= (sin(x))1/x(d)y2=ex+y+ cos(x2)5. Using L’Hˆopital’s Rule, if it is needed, evaluate the following limits.[5 marks](a) limx→0log2(x+1)x(b)limx→∞x2sin(1/x)6. A pile of sand forms a cone (V=1πr2h). Over time, rainfall wears away at the cone,[7 marks]causing the height of the cone to decrease, while maintaining a conical shape. None of