MAT1512 EXAM PACK 2023 LATEST QUESTIONS AND ANSWERS 1 [TURN OVER] 2 x 0 3  x  9 x2  x  QUESTION 1 (a) Determine the following limits (if they exist): (i) (3) (ii) x  2 lim 3  x 1 2 (3) x 3 x  9 (iii) (iv) lim x  2x lim 1  x (3) (3) (v) x

MAT1512 EXAM PACK 2023 LATEST QUESTIONS AND ANSWERS 1 [TURN OVER] 2 x 0 3  x  9 x2  x  QUESTION 1 (a) Determine the following limits (if they exist): (i) (3) (ii) x  2 lim 3  x 1 2 (3) x 3 x  9 (iii) (iv) lim x  2x lim 1  x (3) (3) (v) x  1 lim 1  x 2x (3) (vi) lim sin t  tan 2t (3) t  0 t (b) Use the Squeeze Theorem to determine the following limit: 5k 2  cos3k lim 2 . (3)  x  2 if (c) Let G(x)  x  2 if x  2 x  2 k   k 10 x  2 QUESTION 2 (a) By the first principles of differentiation, find the derivative of f x  2x2  3x  4 at x  2 . (5) (b) Find the derivatives of the following functions by using the appropriate rules of differentiation: (i) f x  (3) sin x cos x (ii) gx  cos5xsinx2  (3)  x  3 lim 6  x  2 x2  4x  2x (i) Draw the graph of Gx. (1) (ii) Determine lim G(x) . (2) (iii) Is Gx continuous at x  2 ? Give a reason for your answer. (1) [25] 3  x  x2  x2  1 dx   (iii) (c) Given: z3 F z   sin 3xdx z sinx  y  2x , find the following: (5) (i) dy by using implicit differentiation