Limts continuity target iit 2013

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Limts continuity target iit 2013

  1. 1. Mission IIT –2013 sec2 x 2 f (t )dtQ1. lim equals x  2 4 x – 2 16 8 2 2 1 (a) f(2) (b) f(2) (c) f  (d) 4 f(2)    2 t 2 f(x) – x 2 f (t )Q2. Let f(x) be differentiable on the interval (0, ) such f(1) = 1, and lim = 1 for t x t–x each x > 0. Then f(x) is 1 2x 2 –1 4x 2 –1 2 1 (a) + (b) + (c) + 2 (d) 3x 3 3x 3 x x x The value of lim   sin x   1  x   , where x > 0 is 1/ x sin xQ3. x 0 (a) 0 (b) –1 (c) 1 (d) 2Q4. If f(x) is continuous and differentiable function and f(1/n) = 0  n  1 and n  I, then (a) f(x) = 0, x  (0, 1] (b) f(0) = 0, f’(0) = 0 (c) f(0) = 0 = f’(0), x  (0, 1] (d) f(0) = 0 and f’(0) need not to be zeroQ5. The function given by y = | | x | – 1| is differentiable for all real numbers except the points (a) {0, 1, –1} (b) 1 (c) 1 (d) –1 f (x 2 ) – f (x)Q6. If (x) is differentiable and strictly increasing function, then the value of lim is x  0 f (x) – f (0) (a) 1 (b) 0 (c) –1 (d) 2 f (2h + 2 + h 2 ) – f (2)Q7. lim , given that f’(2) = 6 and f’(1) = 4 h  0 f (h – h 2  1) – f (1) (a) does not exist (b) is equal to –3/2 (c) is equal to 3/2 (d) is equal to 3  (a – n)nx – tan x  sin nxQ8. If lim = 0, where n is non-zero real number, then a is equal to x 0 x2
  2. 2. Mission IIT –2013 n 1 1 (a) 0 (b) (c) n (d) n+ n n  f 1  x   1/ xQ9. Let f : R R be such that f(1) = 3 and f’(1) = 6. Then lim   equals x 0  f (1)  (a) 1 (b) e1/2 (c) e2 (d) e3 (cos x –1)(cos x – e x )Q10. The integer n for which lim is a finite non-zero number is x 0 xn (a) 1 (b) 2 (c) 3 (d) 4  tan –1 x if |x|  1 Q11. The domain of the derivative of the function  1 is  | x | –1 if x  1 2 (a) R – {0} (b) R – {1} (c) R – {–1} (d) R – {–1, 1}Q12. Which of the following functions is differentiable at x = 0? (a) cos(|x|) + |x| (b) cos(|x|) – |x| (c) sin(|x|) + |x| (d) sin(|x|) – |x|Q13. Let f : R R be a function defined by f(x) = max {x, x3}. The set of all points where f(x) is NOT differentiable is (a) {–1, 1} (b) {–1, 0} (c) {0, 1} (d) {–1, 0, 1}Q14. The left-hand derivative of f(x) = [x] sin( x) at x = k, k an integer, is (a) (–1)k (k – 1) (b) (–1)k – 1(k – 1) (c) (–1)k k (d) (–1)k – 1 k sin( cos 2 x)Q15. lim equals x 0 x2 (a) – (b)  (c) /2 (d) 1 x  x – 3Q16. For x  R, lim   = x 0  x + 2  (a) e (b) e–1 (c) e–5 (d) e5
  3. 3. Mission IIT –2013 x tan 2x – 2x tan xQ17. lim x 0 (1 – cos 2x) 2 (a) 2 (b) –2 (c) 1/2 (d) –1/2Q18. The function f(x) = (x2 – 1) |x2 – 3x + 2| + cos(| x |) is NOT differentiable at (a) –1 (b) 0 (c) 1 (d) 2Q19. The function f(x) = [x]2 – [x2] (where [y] is the greatest integer less than or equal to y), is discontinuous at (a) all integers (b) all integers except 0 and 1 (c) all integers except 0 (d) all integers except 1 1 2n rQ20. lim 1 2 2 equals x  n r= n r (a) 1+ 5 (b) –1 + 5 (c) –1 + 2 (d) 1+ 2

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