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Chapter, , 13, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , LIMITS AND DERIVATIVES, , 13.1 Overview, , 13.1.1 Limits of a function, Let f be a function defined in a domain which we take to be an interval, say, I. We shall, study the concept of limit of f at a point ‘a’ in I., , We say lim– f ( x) is the expected value of f at x = a given the values of f near to the, x →a, , left of a. This value is called the left hand limit of f at a., , f (x) is the expected value of f at x = a given the values of f near to the, We say xlim, →a +, , right of a. This value is called the right hand limit of f at a., , If the right and left hand limits coincide, we call the common value as the limit of f at, , x = a and denote it by lim f ( x) ., x →a, , Some properties of limits, , Let f and g be two functions such that both lim f ( x) and lim g ( x) exist. Then, x →a, , (i) lim [ f ( x) + g ( x)] = lim f ( x) + lim g ( x), x →a, , x→ a, , x→a, , (ii) lim [ f ( x) − g ( x)] = lim f ( x) − lim g ( x), x →a, , x→ a, , x→a, , (iii) For every real number α, , lim ( α f ) ( x) = α lim f ( x), x→a, , no, , x →a, , (iv), , lim [ f ( x) g ( x)] = [lim f ( x) lim g ( x)], , x →a, , lim, , x →a, , x→ a, , x→ a, , f ( x), f ( x) lim, = x →a, g ( x) lim g ( x) , provided g (x) ≠ 0, x→ a, , x →a

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226, , EXEMPLAR PROBLEMS – MATHEMATICS, , Limits of polynomials and rational functions, If f is a polynomial function, then lim f ( x) exists and is given by, x →a, , lim f ( x) = f ( a ), x →a, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , An Important limit, , An important limit which is very useful and used in the sequel is given below:, lim, , x →a, , xn − a n, = na n − 1, x −a, , Remark The above expression remains valid for any rational number provided ‘a’ is, positive., Limits of trigonometric functions, To evaluate the limits of trigonometric functions, we shall make use of the following, limits which are given below:, (i), , lim, , x →0, , sin x, cos x = 1, = 1 (ii) lim, x →0, x, , sin x = 0, (iii) lim, x →0, , 13.1.2 Derivatives Suppose f is a real valued function, then, f ′(x) = lim, h →0, , f ( x + h ) − f ( x), h, , ... (1), , is called the derivative of f at x, provided the limit on the R.H.S. of (1) exists., Algebra of derivative of functions Since the very definition of derivatives involve, limits in a rather direct fashion, we expect the rules of derivatives to follow closely that, of limits as given below:, Let f and g be two functions such that their derivatives are defined in a common, domain. Then :, (i) Derivative of the sum of two function is the sum of the derivatives of the functions., , no, , d, d, d, [ f ( x) + g ( x) ] = f ( x) + g ( x), dx, dx, dx, , (ii) Derivative of the difference of two functions is the difference of the derivatives, of the functions., d, d, d, [ f ( x) − g ( x) ] = f ( x) − g ( x), dx, dx, dx

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LIMITS AND DERIVATIVES, , 227, , (iii) Derivative of the product of two functions is given by the following product, rule., , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , ⎛d, ⎞, ⎛ d, ⎞, d, [ f (x) ⋅ g ( x)] = ⎜⎝ dx f (x )⎟⎠ ⋅ g ( x) + f ( x) ⋅ ⎜⎝ dx g ( x)⎟⎠, dx, This is referred to as Leibnitz Rule for the product of two functions., , (iv) Derivative of quotient of two functions is given by the following quotient rule, (wherever the denominator is non-zero)., ⎛d, ⎞, ⎛d, ⎞, f ( x) ⎟ ⋅ g ( x ) − f ( x) ⋅ ⎜ g ( x)⎟, ⎠, d ⎛ f ( x) ⎞ ⎜⎝ dx, ⎝ dx, ⎠, =, ⎜, ⎟, 2, dx ⎝ g ( x) ⎠, ( g ( x) ), , 13.2 Solved Examples, Short Answer Type, , 2 (2 x − 3) ⎤, ⎡ 1, − 3, Example 1 Evaluate lim ⎢, x →2 x − 2, x − 3 x 2 + 2 x ⎥⎦, ⎣, Solution We have, , 2 (2 x − 3) ⎤, ⎡ 1, lim ⎢, − 3, x, −, 2, x − 3 x 2 + 2 x ⎥⎦ =, ⎣, , x →2, , 2(2 x − 3) ⎤, ⎡ 1, lim ⎢, −, ⎥, ⎣ x − 2 x ( x − 1) (x − 2 ⎦, , x →2, , ⎡ x ( x − 1) − 2 (2 x − 3) ⎤, = xlim, ⎥, →2 ⎢, ⎣ x ( x − 1) ( x − 2) ⎦, , no, , ⎡ x 2 − 5x + 6 ⎤, lim, ⎥, = x →2 ⎢, ⎣ x (x − 1) (x − 2) ⎦, , ⎡ ( x − 2) ( x − 3) ⎤, = lim, [x – 2 ≠ 0], x →2 ⎢ x ( x − 1) ( x − 2) ⎥, ⎣, ⎦, , ⎡ x − 3 ⎤ −1, =, = lim, x →2 ⎢ x ( x − 1) ⎥, ⎣, ⎦ 2

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228, , EXEMPLAR PROBLEMS – MATHEMATICS, , Example 2 Evaluate lim, x →0, , 2+ x− 2, x, , Solution Put y = 2 + x so that when x → 0, y → 2. Then, 1, , 1, , =, , y 2 − 22, lim, y→ 2 y − 2, , =, , 1 2 −1 1 − 2, 1, (2), = ⋅2 =, 2, 2, 2 2, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , 2+ x− 2, , lim, , x →0, , x, , 1, , Example 3 Find the positive integer n so that lim, , x →3, , 1, , x n − 3n, = 108 ., x −3, , Solution We have, , lim, , x →3, , Therefore,, Comparing, we get, , x n − 3n, = n(3)n – 1, x −3, , n(3)n – 1 = 108 = 4 (27) = 4(3)4 – 1, n= 4, , Example 4 Evaluate lim (sec x − tan x), x→, , Solution Put y =, , π, 2, , π, π, − x . Then y → 0 as x → . Therefore, 2, 2, lim (sec x − tan x) = lim [sec(, π, , x→, , y→0, , 2, , π, π, − y ) − tan ( − y)], 2, 2, , no, , (cosec y − cot y ), = lim, y→0, cos y ⎞, ⎛ 1, −, = lim ⎜, y→0 ⎝ sin y, sin y ⎟⎠, ⎛ 1 − cos y ⎞, = lim ⎜, y→0 ⎝, sin y ⎟⎠

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LIMITS AND DERIVATIVES, , 2 sin 2, , = lim, , 2sin, , y 1 − cos y ⎞, ⎛, since , sin 2 =, ⎜, ⎟, 2, 2, ⎜, ⎟, ⎜ sin y = 2sin y cos y, ⎟, ⎝, ⎠, 2, 2, , y, 2, , y, y, cos, 2, 2, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , y →0, , 229, , =, , Example 5 Evaluate lim, x →0, , lim tan, y, →0, 2, , y, =0, 2, , sin (2 + x) − sin(2 − x), x, , Solution (i) We have, , sin (2 + x) − sin(2 − x), lim, =, x→ 0, x, , 2cos, , lim, , x→ 0, , = lim, x →0, , (2 + x + 2 − x ), (2 + x − 2 + x ), sin, 2, 2, x, , 2cos 2 sin x, x, , = 2 cos2 lim, , x →0, , sin x, = 2cos 2, x, , sin x ⎞, ⎛, = 1⎟, ⎜⎝ as xlim, ⎠, →0, x, , Example 6 Find the derivative of f (x) = ax + b, where a and b are non-zero constants,, by first principle., Solution By definition,, f ′(x) = lim, , h →0, , = lim, h →0, , f ( x + h ) − f ( x), h, , bh, a ( x + h) + b − ( ax + b ), = lim, =b, h→ 0 h, h, , no, , Example 7 Find the derivative of f (x) = ax 2 + bx + c, where a, b and c are none-zero, constant, by first principle., Solution By definition,, f ′(x) = lim, , h →0, , f ( x + h ) − f ( x), h

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230, , EXEMPLAR PROBLEMS – MATHEMATICS, , = hlim, →0, , bh + ah 2 + 2axh, = lim, ah + 2ax + b = b + 2ax, h→0, h, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , = lim, h →0, , a ( x + h)2 + b ( x + h) + c − ax 2 − bx − c, h, , Example 8 Find the derivative of f (x) = x3, by first principle., Solution By definition,, f ′(x) = lim, , h →0, , f ( x + h ) − f ( x), h, , = lim, , ( x + h) 3 − x 3, h, , = lim, , x 3 + h 3 + 3 xh ( x + h ) − x3, h, , h →0, , h →0, , = lim, (h 2 + 3x (x + h)) = 3x2, h→0, , Example 9 Find the derivative of f (x) =, , 1, by first principle., x, , Solution By definition,, , f ′(x) =, , lim, h →0, , f (x + h) − f (x), h, , = lim, , 1⎛ 1, 1⎞, − ⎟, ⎜, h ⎝ x + h x⎠, , = lim, h →0, , −h, −1, = 2., h ( x + h) x, x, , h →0, , no, , Example 10 Find the derivative of f (x) = sin x, by first principle., Solution By definition,, f ′(x) =, , lim, h →0, , f (x + h) − f (x), h

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LIMITS AND DERIVATIVES, , = hlim, →0, , 231, , sin ( x + h ) − sin x, h, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , ⎛ 2x + h ⎞, h, 2cos ⎜, sin, ⎝ 2 ⎟⎠, 2, = lim, h →0, h, 2⋅, 2, , h, sin, (2 x + h), 2, ⋅ lim, = lim cos, h →0, h →0, h, 2, 2, , = cos x.1 = cos x, Example 11 Find the derivative of f (x) = xn , where n is positive integer, by first, principle., Solution By definition,, , f ′(x) =, , =, , f ( x + h ) − f ( x), h, , ( x + h ) n − xn, h, , Using Binomial theorem, we have (x + h)n = nC0 xn + nC1 xn – 1 h + ... + n Cn h n, Thus,, , lim, , f ′(x) =, , h →0, , =, , lim, , ( x + h ) n − xn, h, , h ( nx n −1 + ... + h n −1 ], , h →0, , h, , no, , Example 12 Find the derivative of 2x4 + x., Solution Let y = 2x4 + x, , Differentiating both sides with respect to x, we get, dy, =, dx, , d, d, (2 x4 ) +, ( x), dx, dx, , = 2 × 4x4 – 1 + 1x0, , = nx n – 1.

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LIMITS AND DERIVATIVES, , Example 15 Evaluate lim, x →0, , 233, , tan x − sin x, sin 3 x, , Solution We have, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , ⎛ 1, ⎞, −1, sin x ⎜, ⎝ cos x ⎟⎠, , lim, , x →0, , tan x − sin x, = xlim, →0, sin 3 x, = lim, , x →0, , sin 3 x, , 1 − cos x, cos x sin 2 x, , 2 sin 2, , = lim, , x →0, , Example 16 Evaluate lim, , x →a, , Solution We have lim, , x →a, , 3a + x − 2 x, , a + 2x − 3x, , 3a + x − 2 x, , = lim, x →a, , no, , x →a, , (, , 1, ., 2, , a + 2x − 3x, , a + 2x − 3x, , 3a + x − 2 x, , x →a, , lim, , =, , x, x⎞, ⎛, cos x ⎜ 4 sin 2 ⋅ cos 2 ⎟, ⎝, 2, 2⎠, , = lim, , =, , x, 2, , ( a − x), , a + 2 x + 3x, , =, , )(, , (, , (, , x →a, , 3a + x − 2 x, , 3a + x + 2 x, , (, , a + 2 x + 3x, , a + 2 x + 3x, , a + 2x − 3x, , 3a + x − 2 x, , lim, , ×, , )(, , )(, , a + 2x + 3x, , ), , 3a + x + 2 x, , ), , (a − x) ⎡⎣ 3 a + x + 2 x ⎤⎦, a + 2 x + 3 x ( 3a + x − 4 x ), , ), , )

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234, , EXEMPLAR PROBLEMS – MATHEMATICS, , 4 a, 2, 2 3, =, =, ., 3 × 2 3a, 3 3, 9, , =, , Example 17 Evaluate lim, , cos ax − cos bx, cos cx − 1, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , x →0, , Solution We have lim, , (a − b) x, ⎛ (a + b ) ⎞, 2sin ⎜, x⎟ sin, ⎝ 2, ⎠, 2, , x →0, , 2, , sin 2 cx, 2, , 2sin, , =, , lim, x →0, , (a + b ) x, (a − b) x, ⋅ sin, x2, 2, 2, ⋅, cx, x2, sin 2, 2, , ( a + b) x, (a − b ) x, sin, 2, 2, ⋅, = lim, x →0 (a + b ) x ⎛, 2 ⎞ (a − b ) x 2, ⋅, ⋅⎜, a−b, 2, ⎝ a + b ⎟⎠, 2, sin, , =, , Example 18 Evaluate lim, h →0, , Solution We have lim, h →0, , h →0, , = lim [, h →0, , a2 − b2, ⎛a + b a −b 4 ⎞, ×, × 2⎟ =, ⎜⎝, 2, 2, c ⎠, c2, , ( a + h) 2 sin ( a + h ) − a 2 sin a, h, , ( a + h) 2 sin ( a + h ) − a 2 sin a, h, , ( a 2 + h 2 + 2 ah ) [sin a cos h + cos a sin h ] − a 2 sin a, h, , no, = lim, , 2, , 4, ⎛ cx ⎞, ⎜⎝ ⎟⎠ × 2, c, ⋅ 2, 2 cx, sin, 2, , a 2 sin a (cos h − 1) a 2 cos a sin h, +, + ( h + 2 a ) (sin a cos h + cos a sin h )], h, h

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LIMITS AND DERIVATIVES, , 235, , ⎡ 2, ⎤, 2 h, ⎢ a sin a ( −2sin 2 ) h ⎥, a 2 cos a sin h, lim ⎢, ⋅ ⎥ + lim, + lim ( h + 2 a ) sin (a + h ), 2, h →0, h →0, h→0, h, h, 2, =, ⎢, ⎥, ⎥⎦, 2, ⎣⎢, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , = a 2 sin a × 0 + a2 cos a (1) + 2a sin a, , = a2 cos a + 2a sin a., Example 19 Find the derivative of f (x) = tan (ax + b), by first principle., Solution We have f ′(x) = lim, h →0, , = lim, h →0, , f (x + h) − f (x), h, , tan ( a ( x + h) + b ) − tan ( ax + b ), h, , sin ( ax + ah + b) sin ( ax + b ), −, cos ( ax + ah + b ) cos ( ax + b ), = hlim, →0, h, = hlim, →0, , sin ( ax + ah + b ) cos ( ax + b ) − sin ( ax + b ) cos (ax + ah + b ), h cos ( ax + b ) cos ( ax + ah + b ), , = lim, h →0, , a sin ( ah ), a ⋅ h cos ( ax + b ) cos ( ax + ah + b), , = lim, , a, sin ah, lim, [as h → 0 ah → 0], ah, →, 0, cos ( ax + b ) cos ( ax + ah + b ), ah, , h →0, , =, , a, = a sec2 (ax + b)., cos (ax + b), 2, , no, , Example 20 Find the derivative of f ( x) = sin x , by first principle., Solution By definition,, f ′(x) = lim, , h →0, , f ( x + h ) − f ( x), h

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236, , EXEMPLAR PROBLEMS – MATHEMATICS, , sin (x + h) − sin x, h, , = lim, h →0, , = lim, , (, , sin ( x + h ) − sin x, , h →0, , (, , sin ( x + h ) + sin x, , sin ( x + h ) + sin x, , ), , ), , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , h, , )(, , = lim, h →0, , h, , (, , sin ( x + h ) − sin x, , sin ( x + h ) + sin x, , ), , ⎛ 2x + h⎞, h, 2 cos ⎜, sin, ⎝ 2 ⎟⎠, 2, = lim, h →0, h, 2⋅, sin ( x + h) + sin x, 2, , (, , =, , ), , cos x, 1, = cot x sin x, 2 sin x, 2, , Example 21 Find the derivative of, , Solution Let y =, , cos x, ., 1 + sin x, , cos x, 1 + sin x, , Differentiating both sides with respects to x, we get, d ⎛ cos x ⎞, dy, =, dx ⎜⎝ 1 + sin x ⎟⎠, dx, (1 + sin x), , no, , =, , =, , d, d, (cos x) − cos x, (1 + sin x ), dx, dx, (1 + sin x) 2, , (1 + sin x) ( − sin x ) − cos x (cos x), (1 + sin x)2

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LIMITS AND DERIVATIVES, , =, , − (1 + sin x), −1, 2 =, (1 + sin x), 1 + sin x, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , =, , − sin x − sin 2 x − cos 2 x, (1 + sin x) 2, , 237, , Objective Type Questions, Choose the correct answer out of the four options given against each Example 22 to 28, (M.C.Q.)., Example22 lim, x →0, , (A) 0, , sin x, is equal to, (1, x + cos x ), (B), , 1, 2, , (C) 1, , (D) –1, , Solution (B) is the correct answer, we have, sin x, lim, =, x →0 x (1 + cos x ), , x, x, cos, 2, 2, lim, x →0, ⎛, 2 x⎞, x ⎜ 2cos ⎟, ⎝, 2⎠, 2sin, , =, , 1, lim, 2 x→0, , Example23 lim, , π, x→, 2, , 1 − sin x, is equal to, cos x, , (A) 0, (B) –1, Solution (A) is the correct answer, since, , no, , x, 2 = 1, x, 2, 2, , tan, , 1 − sin x, lim, =, π, cos x, x→, 2, , (C) 1, , (D) does not exit, , ⎡, ⎛π, ⎞⎤, ⎢1 − sin ⎜⎝ 2 − y⎟⎠ ⎥ ⎛, π, lim ⎢, ⎥ ⎜ taking − x =, y→0, 2, ⎢ cos ⎛⎜ π − y⎞⎟ ⎥ ⎝, ⎝2, ⎠ ⎥⎦, ⎢⎣, , ⎞, y⎟, ⎠

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238, , EXEMPLAR PROBLEMS – MATHEMATICS, , 2sin 2, , 1 − cos y, lim, = y→0, y, y, sin y, 2 sin cos, 2, 2, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , = lim, y→0, , y, 2, , = lim tan, y→0, , y, =0, 2, , | x|, is equal to, x →0 x, , Example 24 lim, , (A) 1, (B) –1, Solution (D) is the correct answer, since, R.H.S =, , and, , (C) 0, , lim, , | x| x, = =1, x x, , L.H.S =, , lim–, , | x| −x, =, = −1, x, x, , x →0 +, , x →0, , (D) does not exists, , [ x − 1] , where [.] is greatest integer function, is equal to, Example 25 lim, x →1, (A) 1, (B) 2, Solution (D) is the correct answer, since, , (C) 0, , lim [ x − 1] = 0, , R.H.S =, , and, , x →0, , (A) 0, , no, , x →1 +, , L.H.S =, , Example 26 lim x sin, , lim [ x − 1] = –1, , x →1 −, , 1, is equals to, x, (B) 1, , (C), , 1, 2, , Solution (A) is the correct answer, since, lim x = 0 and –1 ≤ sin, x →0, , (D) does not exists, , 1, ≤ 1, by Sandwitch Theorem, we have, x, , (D) does not exist

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LIMITS AND DERIVATIVES, , lim x sin, , x →0, , 1, = 0, x, , 1 + 2 + 3 + ... + n, , n ∈ N, is equal to, n2, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , Example 27 nlim, →∞, , 239, , (A) 0, , (B) 1, , (C), , Solution (C) is the correct answer. As xlim, →∞, = nlim, →∞, , n ( n + 1), =, 2n2, , 1, 2, , (D), , 1, 4, , (D), , 1, 2, , 1 + 2 + 3 + ... + n, n2, , lim, , x →∞, , 1⎛, ⎜ 1+, 2⎝, , 1⎞ 1, ⎟=, n⎠ 2, , ⎛ π⎞, Example 28 If f(x) = x sinx, then f ′ ⎜⎝ ⎟⎠ is equal to, 2, (A) 0, , (B) 1, , (C) –1, , Solution (B) is the correct answer. As f′ (x) = x cosx + sinx, , ⎛ π⎞, f ′⎜ ⎟ =, ⎝ 2⎠, , So,, , π, π, π, cos + sin = 1, 2, 2, 2, , 13.3 EXERCISE, , Short Answer Type, Evaluate :, x2 − 9, x −3, , no, , 1. lim, x →3, , 2., , 1, , lim, , x→, , 1, 2, , 4x 2 − 1, 2x − 1, , 1, , ( x + 2) 3 − 2 3, 4. lim, x →0, x, , 3. lim, h →0, , x+ h − x, h, , 5, , (1 + x) 6 − 1, 5. lim, x→1 (1 + x) 2 − 1, , 5, , (2 + x) 2 − ( a + 2) 2, 6. lim, x →a, x−a

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LIMITS AND DERIVATIVES, , 32. (sec x – 1) (sec x + 1) 33., , x 2 cos, , 34., , x5 − cos x, sin x, , 36. (ax2 + cotx) (p + q cosx), , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , 35., , π, 4, , 3x + 4, 2, 5x − 7 x + 9, , 241, , 37., , sin x, , a + b sin x, c + d cos x, , 40. x2 sinx + cos2x, , 38. (sin x + cosx) 2, , 39. (2x – 7)2 (3x + 5)3, , 41. sin3x cos3x, , 42., , 1, ax + bx + c, 2, , Long Answer Type, Differentiate each of the functions with respect to ‘x’ in Exercises 43 to 46 using first, principle., 43. cos (x2 + 1), , 44., , ax + b, cx + d, , 2, , 45. x 3, , 46. x cosx, , Evaluate each of the following limits in Exercises 47 to 53., 47. lim, y→0, , ( x + y) sec(x + y) − x sec x, y, , 48. xlim, →0, , (sin( α + β) x + sin(α − β) x + sin 2 α x), ⋅x, cos 2βx − cos 2αx, , 49. limπ, , no, , x→, , tan 3 x − tan x, π⎞, ⎛, cos ⎜ x + ⎟, ⎝, 4⎠, , 4, , 51. Show that lim, x →4, , 1 − sin, , 50., , lim, , x →π, , x, 2, , x⎛, x, x⎞, cos ⎜ cos − sin ⎟, 2⎝, 4, 4⎠, , | x − 4|, does not exists, x −4

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242, , EXEMPLAR PROBLEMS – MATHEMATICS, , π, 2, π, and if lim f ( x) = f ( ) ,, π, 2, π, x→, 2, x=, 2, , ⎧ k cos x, ⎪⎪ π − 2 x, 52. Let f(x) = ⎨, ⎪ 3, ⎪⎩, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , when x ≠, , find the value of k., , ⎧x + 2, 53. Let f (x) = ⎨ 2, ⎩ cx, , x ≤ –1, , x > −1, , f ( x) exists., , find ‘c’ if xlim, →–1, , Objective Type Questions, Choose the correct answer out of 4 options given against each Exercise 54 to 76, (M.C.Q)., 54., , lim, , x →π, , sin x, is, x −π, , (A) 1, , 55., , (B), , 3, 2, , (1 + x) n − 1, is, x →0, x, (A) n, (B) 1, , −3, 2, , (D) 1, , (C) –n, , (D) 0, , (C), , lim, , lim, x→1, , xm − 1, is, xn − 1, , (A) 1, , 58., , (D) –2, , x2 cos x, is, x →0 1 − cos x, , no, , 57., , (C) –1, , lim, , (A) 2, , 56., , (B) 2, , 1 − cos4 θ, is, x →0 1 − cos6 θ, lim, , (B), , m, n, , (C) −, , m, n, , (D), , m2, n2

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LIMITS AND DERIVATIVES, , 4, 9, , (A), , (B), , 1, 2, , (C), , −1, 2, , 243, , (D) –1, , cosec x − cot x, is, x, , 59. xlim, →0, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , −1, 2, , (A), , (C), , 1, 2, , (D) 1, , sin x, is, x + 1 − 1− x, , 60. xlim, →0, , (A) 2, , 61., , (B) 1, , (B) 0, , (C) 1, , (D) –1, , (B) 1, , (C) 0, , (D), , (C) 1, , (D) None of these, , sec2 x − 2, is, π tan x − 1, x→, lim, 4, , (A) 3, , 62. lim, , (, , x − 1) ( 2 x − 3), 2 x2 + x − 3, , x→1, , (A), , 1, 10, , 2, , is, , (B), , −1, 10, , ⎧ sin[ x], , [ x] ≠ 0, ⎪, , where [.] denotes the greatest integer function ,, 63. If f (x) = ⎨ [ x], ⎪ 0 ,[ x ] = 0, ⎩, f ( x) is equal to, then lim, x →0, , (A) 1, , (B) 0, , (C) –1, , (D) None of these, , | sin x |, is, x, (A) 1, , (B) –1, , (C) does not exist(D) None of these, , 64. lim, , no, , x →0, , ⎧ x2 − 1, 0 < x < 2, f ( x) and, , the quadratic equation whose roots are xlim, 65. Let f (x) = ⎨, →2 –, ⎩ 2 x + 3, 2 ≤ x < 3, lim f ( x) is, , x →2 +

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244, , EXEMPLAR PROBLEMS – MATHEMATICS, , (A) x2 – 6x + 9 = 0, (C) x2 – 14x + 49 = 0, tan 2 x − x, is, x →0 3 x − sin x, lim, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , 66., , (B) x2 – 7x + 8 = 0, (D) x2 – 10x + 21 = 0, , (A) 2, , (B), , 1, 2, , (C), , −1, 2, , (D), , 1, 4, , ⎛1⎞, 67. Let f (x) = x – [x]; ∈ R, then f ′ ⎜⎝ ⎟⎠ is, 2, , (A), , 3, 2, , (B) 1, , x+, , 68. If y =, , (A), , (D) –1, , dy, 1, , then, at x = 1 is, dx, x, , (A) 1, , (B), , 69. If f (x) =, , (C) 0, , 1, 2, , 1, 2, , (C), , (D) 0, , x−4, , then f ′(1) is, 2 x, , 5, 4, , (B), , 4, 5, , (C) 1, , (D) 0, , 1, dy, x2, 70. If y =, , then, is, 1, dx, 1− 2, x, 1+, , − 4x, 2, (x − 1)2, , no, , (A), , 71. If y =, , (B), , −4 x, x2 −1, , sin x + cos x, dy, , then, at x = 0 is, sin x − cos x, dx, , (C), , 1 − x2, 4x, , (D), , 4x, x −1, 2

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LIMITS AND DERIVATIVES, , (A) –2, , (C), , 1, 2, , (D) does not exist, , sin( x + 9), dy, , then, at x = 0 is, cos x, dx, , tt ©, o N, be C, re ER, pu T, bl, is, he, d, , 72. If y =, , (B) 0, , 245, , (A) cos 9, , (B) sin 9, , 73. If f (x) = 1 + x +, , (A), , x2, 2, , 1, 100, , 74. If f ( x) =, , + ... +, , x100, , 100, , (C) 0, , (D) 1, , , then f ′(1) is equal to, , (B) 100, , (C) does not exist, , (D) 0, , xn − a n, for some constant ‘a’, then f ′(a) is, x− a, , (A) 1, , (B) 0, , (C) does not exist, , 75. If f (x) = x100 + x99 + ... + x + 1, then f ′(1) is equal to, (A) 5050, (B) 5049, (C) 5051, , (D), , 1, 2, , (D) 50051, , 76. If f (x) = 1 – x + x – x ... – x + x , then f ′(1) is euqal to, (A) 150, (B) –50, (C) –150, (D) 50, Fill in the blanks in Exercises 77 to 80., 2, , 77. If f (x) =, , 3, , 99, , 100, , tan x, f ( x) = ______________, , then lim, x →π, x−π, , x ⎞, 78. lim ⎛⎜ sin mx cot, ⎟ = 2 , then m = ______________, x →0 ⎝, 3⎠, , no, , 79. if y = 1 +, , 80., , lim, , x →3+, , dy, x x2 x 3, +, +, + ... , then, = ______________, dx, 1! 2! 3!, , x, = ______________, [ x]