Tiêu đề 02A. LIMITS, CONTINUITY & DIFFERENTIABILITY FINAL ( 49-90).pdf ✅

NISHITH Multimedia India (Pvt.) Ltd., 4 9 JEE MAINS - CW - VOL - I LIMITS, CONTINUITY & DIFFERENTIABILITY NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL - III LIMITS, CONTINUITY & DIFFERENTIABILITY SYNOPSIS THEOREMS ON CONTINUITY (i) If f x and g x     are continuous at x a  , then the following functions are continuous at x a  : (a) f x g x      (b) kf x k R  ,  (c) kg x k R  ,  (d) f x g x  .   (e)     , f x g x provided g a   0 (f)   p q/ f x (provided    p q/ f x is defined on an interval containing a,p and q are integers) (ii) If f x  is continuous and g x  is discontinuous at x a  , then (a) f x g x      is discontinuous (b) f x g x  .   may be continuous or discontinuous. (iii) If f x  is discontinuous and g x  is discontinuous at x a  , then (a) f x g x      may be continuous or discontinuous. (b) f x g x  .   may be continuous or discontinuous. (iv) Every polynomial as continuous at every point of the real line. Suppose f x  is a nth degree polynomial function.   1 2 0 1 2 ........... n n n n f x a x a x a x a x R           (v) Every rational function is continuous at every point where its denominator is different from zero. (vi) Logarithmic functions, Exponential functions, Trigonometric functions, Inverse circular functions, Modulus functions are continuous in their domain. (vii) Point function (domain and range consist of one value only) is discontinuous function. e.g., f x x x        2 2    discontinuous at x  2.  in x x   2, 2 (LHL is not defined) and in 2 , 2   x x (RHL is not defined)  f 2 0   THEOREMS ON DIFFERENTIABILITY (i) If f x  and g x  are differentable at 0 x x  then the functions f x g x f x g x     , .     will be also be differentiable at 0 x x  and also       0  0 f x g x g x  is differentable at 0 x x  . (ii) If f x  is differentiable and g x  is non- differentiable at 0 x x  , then f x g x     is non-differentiable and f x g x  .   may be differentaible or non-differentiable. (iii) If f x  and g x  both are non-differentiable at 0 x x  then f x g x f x g x     , .     may be differentiable or non-differentiable at 0 x x  .
5 0 NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL - III NISHITH Multimedia India (Pvt.) Ltd., LIMITS, CONTINUITY & DIFFERENTIABILITY (x) The trigonometric functions sin ,cos , tan ,cot ,sec ,cos x x x x x ecx are differentiable in their domain. (viii) The exponential functions , 0 1   x x e a a and a   are differentiable at each x R  . (ix) The logarthmic functions in ,log 1   a x x a o and a   are differentiable at each point in their domain. (xi) The inverse trigonometric functions 1 1 1 1 1 1 sin ,cos , tan ,cot ,sec ,cos x x x x x ec x       are differentiable in their domain. (xii) The composition of differentiable functions is also differentiable function. (iv) If f x  and g x  are inverse function of each other then      / / 1 g x f g x  and      / / 1 f x g f x  where          / . df g x f g x d g x  (v) Derivative of an identity function is an identity function. (vi) Every polynomial function is differentiable at each x R  . (vii) Every constant function is differentiable at each x R  . Remembering method f x  g x  f x g x      f x g x  .   Differentiable Differentiable Differentiable Differentiable Differentiable Non- differentiable Non- differentiable May be differentiable or non-differentiable Non-differentiable Non-differentiable May be differentiable or non-differentiable May be differentiable or non-differentiable Remembering method f x  g x  f x g x     f x g x  .   f x g x  /   provided g x   0 Continuous continuous Continuous Continuous Continuous Continuous discontinuous discontinuous May be continuous or discontinuous May be continuous or discontinuous discontinuous discontinuous May be continuous or discontinuous May be continuous or discontinuous May be continuous or discontinuous
NISHITH Multimedia India (Pvt.) Ltd., 5 1 JEE MAINS - CW - VOL - I LIMITS, CONTINUITY & DIFFERENTIABILITY NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL - III LEVEL - V SINGLE ANSWER QUESTIONS 1. If i i i x a A , i 1, 2,3,...., n x a     and 1 2 3 n a a a ..... a     , then   m 1 2 n x a lim A A ....A ,1 m n    (A) is equal to   m 1 (B) is equal to   m 1 1   (C) is equal to   m 1 1   (D) Does not exist 2. The value of               1/2 1/3 1/4 1/n 1/2 1/3 1/n x 2 x 3 x 4 x ...... n x lim 2x 3 2x 3 ...... 2x 3            is (A) 2 (B)2 (C) 1 3 (D) 0 3. ABC is an isosceles triangle inscribed in a circle of radius r. If AB = AC and h is the altitude from A to BC, then the triangle ABC has perimeter P =   2 2 2 2 hr h hr         and area  then 3 h 0 A Lim P (A) 1 r (B) 1 64r (C) 1 128r (D) 1 2r 4. If         x x x f x ... 1 x x 1 2x 1 2x 1 3x 1           , then at x = 0, f(x) (A) has no limit (B) is discontinuous (C) is continuous but not differentiable (D) is differentiable 5. The integral value of n for which 3 2 0 cos cos cos 2 lim x x n x x x x e x e  x           is finite and non zero is (A) 2 (B) 4 (C) 5 (D) 6 6. The value of 3 3 3 lim ... x x x x x x x x                is (A) 1 B) 0 (C) 2 (D) 1 2 7. If   3 4 2 0 lim 1 cos2 cos3 cos4 ..... cos / n x x x x nx x       is equal to 10, then the value of n is (A) 5 (B) 4 (C) 6 (D) 3 8. Let 1 2 , ,...., n a a a be sequence of real numbers with 2 1 1 n n n a a a     and a1  0 . Then 1 lim 2 n n n a          (A) 2  (B) 2   (C) 2   (D) 2  9. If   3 2 0 lim sin 3 x x x ax b      exists and is equal to 0, then (A) a = –3 and 9 2 b  (B) a =3 and 9 2 b  (C) a = –3 and 9 2 b   (D)a = 3 and 9 2 b   10. A discontinuous function y = f(x) satisfying x 2 + y2 = 4 is given by (A) f(x) = 2 2 4 x 2 x 0 4 x 0 x 2            (B)f(x) = 2 2 4 x 2 x 0 4 x 0 x 2            (C) f(x) = 2 4 x 2 x 2     (D) f(x) = 2      4 x 2 x 2
5 2 NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL - III NISHITH Multimedia India (Pvt.) Ltd., LIMITS, CONTINUITY & DIFFERENTIABILITY 11. If        x x , 0 x 2 f x x 1 x , 2 x 3           , where [.] denotes the greatest integer function, then (A) both f 1   and f 2   do not exist (B) f 1   exists but f 2   does not exist (C) f 2   exists but f 1   does not exist (D) both f 1   and f 2   exists 12. The function f(x) = x sin , x 1 2 2 x 3 [x] , x 1                (A) is continuous at x = 1 (B) is differentiable at x = 1 (C) is continuous but not differentiable at x = 1 (D) x 1 lim f (x)  does not exist 13. Let the function f defined by f(x) = x[x] , 0 x 2 (x 1) [x] , 2 x 3         (where, [x] = greatest integer less than or equal to x). Then f(x) is (A) Continuous but not differentiable at x = 1 (B) Differentiable at x = 1 (C) Continuous but not differentiable at x = 2 (D) Differentiable at x = 2 14. For the function f(x) =        0, x 0 , x 0 1 e x 1/ x , the derivative from the right,f(0) .............. and the derivative from the left, f 0   are (A) {0, 1} (B) {1, 0} (C) {1, 1} (D) {0, 0} 15. Let f 0, R 2         be a function defined by f(x) = 3 max sin x, cos x, 4       , then number of points where f(x) is non differentiable is (A) 1 (B) 2 (C) 3 (D) 0 16. Let f(x) =         ax b for | x | 1 for | x | 1 | x | 1 2 . If f(x) is continuous and differentiable everywhere, then (A) a = 2 1 , b = – 2 3 (B) a = – 2 1 , b = 2 3 (C) a = 1, b = –1 (D) a = b = 1 17 Let f x 3 2cos x , x ,     2 2            , where [.] denotes the greatest integer function. Then number of pooints of discontinuity of f x  is (A) 3 (B) 2 (C) 5 (D) 6 18. Let f x  be a real function not identically zero in R, such that        2n 1 2n 1 f x y f x f y , n N       and x, y R  . If f 0 0    , then f 6   is equal to (A) 0 (B) 1 (C) –1 (D) 2 19. A function f : R  R satisfies the equation f(x) f(y) – f(xy) = x + y  x, y  R and f(1) > 0, then (A) 1 2 f(x) f (x) x 4    (B) 1 2 f(x) f (x) x 6    (C) 1 2 f(x) f (x) x 1    (D) 1 2 f(x) f (x) x 6    20. If f(x) = 2 2 x 1 x 1   , for every real number x, then the minimum value of [IIT - 1998] (A) does not exist because f is unbounded (B) is not attained even though f is bounded (C) is equal to 1 (D) is equal to – 1 21. The function f(x) = [x]2 - [x2 ] (where [x] is the greatest integer less than or equal to x), is discontinuous at : [IIT - 1999] (A) all integers (B) all integers except 0 and 1 (C) all integers except 0 (D) all integers except 1