MATH 31A Midterm: Math 31A Exam 3
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Practice for midterm i (solutions: (a) find lim t 0 (cid:18) Let us start by simplifying what is inside the limit. This will be done by adding fractions, and then multiplying by the conjugate. 3 9 + 2t sin(3t)(3 + 9 + 2t) So applying basic rules of limits we have (cid:18) 3t (cid:19) lim sin(3t) t 0 (3 + 9 + 2t) t 0. Again we start by simplifying what is inside the limit. we have. The function will be continuous at 0 if lim x 0 g(x) = g(0). From the de nition it is easy to see that g(0) = 3. Since this is a piecewise function it is best to split the limit into two parts (from the left and right). If they both are 3 then the function is continuous. Let us attempt to calculate what g (0) should be. We can do this by using the de nition of the derivative.
