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Limits & Derivatives

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Published in: IIT JEE Mains
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  1. Mathematics lim 1 10 DPP cw)Competition World x-3X Peekazea Koc Sgeeeea Limits, Indeterminate Forms - 1 is equal to 1 (b) 10 is equal to lim 1 2 lim( X 00 n 00 a — UI) is equal to , where a > b > 1, is equal to 3x-1 is equal to lim (a) loge 9 is equal to 31/n _ 1 (a) log4 3 cosx+ sin x (b) loge 3 lim X 00 is equal to 2 x is equal to 1 (c) 2 (C) 00 (d) none of these (d) none of these (d) none of these (d) none of these (d) none of these (d) none of these (d) none of these
  2. 1 Q9. Q 10. Q 11. Q 12. x tan2x —2x tan x Q 13. Q 14. Q 15. lim 1 (a) 12 nP sin2(n!) 1 is equal to 4 (b) 3 , 0 < p < 1, is equal to lim lim 1 (a) 4 lim 8xy — 2 x (b) 00 is equal to 8xy 1 (b) 2 is equal to lim (1 — cos loge lim 2 x 1 2 Let sn 1 (a) 3 x 1 1.4 4.7 7.10 is equal to is equal to 1 2 1 + to n terms. Then 16 (c) 3 1 2Uä 1 (c) 2 1 2 lim sn is equal to 1 (c) 4 1 (d) 48 (d) none of these (d) none of these (d) none of these 1 2 (d) none of these (d) 00
  3. DPP Mathematics CW Competition World Peekazea Koc Sgeeeea Limits, Indeterminate Forms - 2 Let the rth term, tr, of a series is given by tr . Then limy tr is n —..>cxo r=l (d) none of these 1 (a) 4 Let a = min {x 2+1 _ 1 (a) 3.2 1 (c) 2 I-cose xe R} and b = 1m 2+1 + I (b) 3.2 n(n + 1) (b) 2 . The value of Ear.bn r is 02 x r=l lim n (a) 2 is equal to x — tan x is equal to lim x 00 x t a n x x—sinx is equal to lim x + sin x loge cosx is equal to lim 2 x 417+1 _ 1 (c) 3.2 1 (c) 2 (C) 00 1 (a) 2 sin lim 1 (a) 2 x — tan 2 x 1 (b) 2 x is equal to 1 2 (d) none of these (d) none of these (d) none of these (d) none of these (d) none of these (d) 00
  4. 1 — cosx is equal to lim (a) —loc e If f(4) = f, f'(4) = (b) —loge 2 1 then lim is equal to Q9. Q 10. Q 11. Q 12. (a) — g(a)f(x) Q 13. x-a Q 14. 2) Q 15. 2—07 The graph of the function y = f(x) has a unique tangent at the point (a, 0) through which the loge{l + 6f(x)} . graph passes. Then lim IS Let f(x) be a twice-differentiable function and = 2 then (c) 12 lim (d) none of these (d) none of these (d) none of these 2f(x) - 3f(2x) + f(4x) is x (d) none of these If f(x), g(x) be differentiable functions and f(l) = g(l) = 2 then lim f(l)g(x) — f(x)g(l) — g(x) —f(x) is equal to If f(a) = 2, f'(a) = 1, g(a) = -1, g'(a) = 2 then 1 5 is equal to lim 1+sin— n 00 lim is (a) ea/2 lim l/x (b) el/ 2 (d) none of these (d) none of these (d) e2a (d) none of these
  5. Mathematics lim tan DPP cw)Competition World lim (2—tanx) Peekazea Koc Sgeeeea Limits, Indeterminate Forms - 3 is equal to lim If lim sinVÖ log tan x is equal to is equal to then (c) X = 1, g = any real constant is equal to lim sine Let {x} denote the fractional part of x. loge [x] Then lim is equal to , where [.] denotes the greatest integer function, is lim x X 00 1 — cos2(x —1) lim X-I (d) none of these (d) none of these (d) none of these (d) nonexistent (a) exists and it is (c) does not exist because x— 1 +0 (b) exists and it is -u (d) does not exist because LH lim RH lim
  6. Q9. Q 10. Q 11. Q 12. Q 13. Q 14. Q 15. x sin{x — lim x[x] , where [.] denotes the greatest integer function, is (c) not existent lim , where [.] denotes the greatest integer function, is (a) is O (b) is 3 (c) not existent (c) is -3 lim {[x]+ I x l} , where [.] denotes the greatest integer function, (a) is 0 Let f(x) = (b) is 1 The quadratic equation whose roots are (c) does not exist lim f(x)and lim f(x) is (c) k - 14 (b) k - + 21 = o If [.] denotes the greatest integer function then lim n 00 x (c) 2 n cos t2dt is equal to lim (d) none of these (d) none of these (d) does not exist (d) none of these (d) none of these is x 2 (d) none of these
  7. DPP cw)Competition World Mathematics Peekazea Koc Sgeeeea Limits, Indeterminate Forms - 4 x lim (a) f(a) is equal to a (b) a.f(a) is equal to lim x. 1+0 x — I ) Iff(x) continuous in [0, 1] and f | then lim f n 00 n 9n2 1 3 (2) — — then If f(x) is continuous and f limf 1 — cos3x x is is equal to 9 (a) 2 2 (b) 9 (d) none of these (d) none of these (d) none of these (d) none of these One or more options may be correct. Let f(x) = 1+ a = ax, 1 SX
  8. If lim an— n 00 (a)a=l = b , a finite number, then Q9. Let tan 01 . x + sin . y = 01 and u cosec . x + cos 01 . y = 1 be two variable straight lines, 01 being the parameter. Let P be the point of intersection of the lines. In the limiting position when 01 + 0, the point P lies on the line (a)x=2

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