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SYNOPSIS ON LIMITS _, DERIVATIVES AND, MATHEMATICAL INDUCTION, , > Limit of a function: If f(x) takes the value nearer to | when x takes the value nearer to a. Then 1 is called, limit of a function and it is denoted by lim f(x) = 1, xa, , , , If x approaches “a” from the left side ie. a— 6 < x <a, then the limit is called left hand limit [LHL] and, denoted as lim_ f(x), xa, , If x approaches “a” from the right side ie. a < x < a+ 6, then the limit is called right hand limit [RHL], and denoted as lim, f(x), xa, , If the LHL and RHL exist and are equal then the function is said to have a limit., > Properties Of Limits: Iflim f(x) =m andlim g(x) =n then, xa xa, * lim [f(x) + g()] =mtn, xa, % lim [f(x).g@)]=mn, xoa, lim [cf(x)] = lim f(x) = cm where c is a constant, xa xa, * lim log, f(x) =log,m, xa, lim [f(x)]9 =m", xa, , > Important Formula:, , , , , , , , , , , , * lim = na"~* ¥ rational values of n, , xoa xo, + lim = = 1 where @ is in radian. lim fen? — 1 where @ is in radian., , 1 x, , + lim +x) =e OR lim (1+2) =e, , x30 xo x, lim =log,a ie lim “* = log, e = 1. im 2820+) = 1, , x90 x Be x90 x Be x00, , > L’Hospital’s rule for calculating limits :, , “Step 1 : first reduce the function to 5 form., , “+ Step 2 : apply the limits, , “Step 3 : after applying limits if we get indeterminate form then differentiate the numerator and, denominator separately w.r.t independent variable and apply the limit, , Step 4 : If we get finite unique quantity as the answer, that itself is its limit otherwise repeat the, step 3, , ‘Indeterminate forms : The forms ¢,~ ,0 x c0,co—,0°,1%and «0° are called indeterminate, forms., , DIFFERENTIATION, > Differentiability of a function : F(x) is differentiable at x = a, if left hand derivative at x =a is equal to, , right hand derivative x = a., , lip FO = FO yy MAD FO _ ay, , h>0- h hoo, , fl h)— fi i, fla +W= FO. 5%, , > Some useful results of differentiations:, d d d d d, DG) =1 2); (constant) = 0 utr) =2W +20), , ice,. lim, hoo

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11.The derivative of f(x)=x|x| is, , , , , , a) |x|+x b)2x c)2|x] d)-2|x|, , 12.1 f(x)}= 2 then f? (=, , a)4+V3 b)4+2V3 c)4(2+v3) d)2+v3, , 13. li (2x~3)(x~4) _, , a)2/5 b)O c)oo d)3/10, , 14.The derivative of f(x)=cos2x is, , a)sin2x b)-sin2x c)cos2x d)sin2x, (igs 1.4)=, , 15.if fx)=(Vx x) then f {-1)=, , a)-1 b)-2 c)O d)2, , x?-1 0<x<2, , ele f= 03 2209, , the quadratic equation whose roots are lim, f(x) and lim,, f(x) is, x x, , , , a)x*?-6x+9=0 b)x?-7x+8=0, c)x? -14x+ 49 =0 d)x?-10x+21=0, ee i Lies, 17.If y=14— HGH GH oe cree ene ee .. 00,With |x|>1, then i, e y2 2, a)x’y? bh a de>, 18.1 f(x)=(x-1)(x-2)(x-3) then f(0)=, a)o b)1 co) de, =" 1(Q)=, 19.1f flx)= then f*(0):, ajo b)1 c)doesnot exists d)none of these, 20.1If f(a)=2 ,f"(a)=1,g(a)=-1,g1(a)=2,then the value of lim gore -sOre., xoa =, a)O b)1 c)doesnot exists d)none of these, 21. lim “is, x90 %, 7 cs « 180, a) b)> hee dj, . cosS0-cos76 ;, 22. lim —a 5s, 1 4, a)= b)S c) 12 d)6, 23, lim 3tame=8 j, x0 3xtsin2x, a) -1 b)1 c)0 d) none of these, 24, tim G42""1 ig, x90 x, a)n b)1 c)—n do, 25. If [x] is the gratest integer function, then lim [x] is, xo, a)O b) 2 c)-1 d) does not exist, 26. Let P(n):2"<n!,where n is a natural number,then P(n) is true for, a)alln bjall n>2 c)all n>3 d)none of these, , 27.Let T(k) be the statement 1+3+5+.............+(2k-1)=k?+10.which of the following is true?, a)T(1) istrue b)T(k)true=>T(k+1) is true c)T(n)istrue VneN _—_d)all the above are correct, 28.If n is a positive integer ,then 2.47"'1433"1 js divisible by, a)2 b)7 c)11 d)27