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This PowerPoint helps students to consider the concept of infinity.
Limts continuity target iit 2013
1. Mission IIT –2013
sec2 x
2
f (t )dt
Q1. 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 x
Q3.
x 0
(a) 0 (b) –1 (c) 1 (d) 2
Q4. 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 zero
Q5. 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 nx
Q8. If lim = 0, where n is non-zero real number, then a is equal to
x 0 x2
2. Mission IIT –2013
n 1 1
(a) 0 (b) (c) n (d) n+
n n
f 1 x
1/ x
Q9. 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. Mission IIT –2013
x tan 2x – 2x tan x
Q17. lim
x 0 (1 – cos 2x) 2
(a) 2 (b) –2 (c) 1/2 (d) –1/2
Q18. The function f(x) = (x2 – 1) |x2 – 3x + 2| + cos(| x |) is NOT differentiable at
(a) –1 (b) 0 (c) 1 (d) 2
Q19. 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 r
Q20. lim 1 2 2 equals
x n
r= n r
(a) 1+ 5 (b) –1 + 5 (c) –1 + 2 (d) 1+ 2