Q: Let x_0 be the real number such that e^ x_0 +x_0=0. For a given real number , define g(x)= 3 x e^x+3 x- e^x- x 3 (e^x+1 ) for all real numbers x. Then which one of the following statements is TRUE?

aligned & e^ x_0 +x_0=0 \\ & g(x)= 3 x e^x+3 x- e^x- x 3 (e^x+1 ) \\ & g(x)=x- (e^x+x ) 3 (e^x+1 ) , g(x)+e^ x_0 =x+e^ x_0 - 3 ( e^x+x e^x+1 ) \\ & g(x)+e^ x_0 = (x-x_0 )- 3 ( e^x+x e^x+1 ) \\ & As very clear g (x_0 )+x_0 0 aligned aligned & _ x x_0 | g(x)+e^ x_0 x-x_0 | 0 0 = _ x x_0 | g^ (x) 1 |= |g^ (x_0 ) | \\ & g^ (x)=1- 3 ( (e^x+1 ) (e^x+1 )- (e^x+x ) e^x (e^x+1 )^2 ) \\ & g^ (x_0 )=1- 3 ( (e^ x_0 +1 )^2 (e^ x_0 +1 )^2 )=1- 3 \\ & _ x x_0 | g(x)+e^ x_0 x-x_0 |= |g^ (x_0 ) |= |1- 3 | aligne

Q: Let k R . If _ x 0^+ ( ( k x)+ x+x)^ 2 x =e^6, then the value of k is

$ aligned & l= _ x 0^ + ( ( k x)+ x+x)^ 2 x =e^6 \\ & l= _ x 0^ + 2 x ( ( k x)+ x+x-1) \\ & l= _ x 0^ + 2 ( ( k x) k x k x k x k x x +1- (1- x) x^2 x ) \\ & l=2(k+1) \\ & l=e^ 2(k+1) =e^6 \\ & k+1=3 \\ & k=2 aligned $

Q: Let S be the set of all ( , ) R R such that _ x (x^2 ) ( _e x )^ ( 1 x^2 ) x^ ( _e(1+x) )^ =0. Then which of the following is (are) correct?

aligned & _ x x^2 ( _e x )^ 1 x^2 x^ ( _e(1+x) )^ =0 \\ & _ x ( _e x )^ ( _e(x+1) )^ x^ +2 =0 aligned aligned & _ x ( _e x _e(x+1) )^ ( _e x )^ - x^ +2 =0 \\ & _ x ( _e x )^ - x^ +2 =0 Put _e x=t \\ & _ t t^ - (e^t )^ +2 =0 aligned As we know _ x x e^x =0 +2>0 >-2

Q: Let the function f:[1, ) R be defined by f(t)= \ array cc (-1)^ n+1 2, & if t=2 n-1, n N , \\ (2 n+1-t) 2 f(2 n-1)+ (t-(2 n-1)) 2 f(2 n+1), & if 2 n-1 < t < 2 n+1, n N . array . Define g(x)= _1^x f(t)

f ( t )= \ array ccc 2 & ; & t =1 \\ 4-2 t & ; & 1

Q: Let f(x) be a continuously differentiable function on the interval (0, ) such that f(1)=2 and t x t^ 10 f(x)-x^ 10 f(t) t^9-x^9 =1 for each x>0. Then, for all x>0, f(x) is equal to

aligned & _ t x t^ 10 f(x)-x^ 10 f(t) t^9-x^9 =1 \\ & _ t x 10 t^9 f(x)-x^ 10 f^ (t) 9 t^8 =1 aligned aligned & 10 x f(x)-x^2 f^ (x)=9 \\ & x^2 f^ (x)=10 x f(x)-9 \\ & f^ (x)= 10 f(x) x - 9 x^2 \\ & d y d x - 10 x y=- 9 x^2 \\ & y 1 x^ 10 = - 9 x^2 1 x^ 10 d x \\ & y x^ 10 = 9 11 x^ 11 +c...(1) aligned aligned & f (1)=2 2 1 = 9 11 + c c = 13 11 \\ & f ( x )= 9 11 x + 13 11 x ^ 10 aligned Option (2) is correct.

Q: Let f : 0 , 1 → ℝ be the function defined as f x = n if x ∈ [ 1 n + 1 , 1 n ) where n ∈ ℕ . Let g : 0 , 1 → ℝ be a function such that &#8

Given, f : 0 , 1 → ℝ be the function defined as f x = n if x ∈ [ 1 n + 1 , 1 n ) where n ∈ ℕ And g : 0 , 1 → ℝ be a function such that ∫ x 2 x 1 - t t d t < g x < 2 x for all x ∈ 0 , 1 . Now we need to solve 1 sided limit here to get some answer, as lim x → 0 - doesn’t exist here (not in domain) As 1 n + 1 ≤ x < 1 n ⇒ n + 1 ≥ 1 x &

Q: If β = lim x → 0 e x 3 - 1 - x 3 1 3 + 1 - x 2 1 2 - 1 sin x x sin 2 x , then the value of 6 β is

β = lim x → 0 e x 3 - 1 - x 3 1 3 + 1 - x 2 1 2 - 1 sin x x · sin 2 x x 2 · x 2 β = lim x → 0 1 + x 3 + x 6 2 ! + ⋯ - 1 - 1 3 x 3 + - 1 9 x 6 + ⋯ + - 1 2 x 2 - 1 8 x 4 + ⋯ x - x 3 3 ! + ⋯ x 3 β = lim x → 0 x 3 1 + 1 3 - 1 2 x 3 (Neglecting higher powers of x ) β = 5 6 i.e. 6 β = 5

Q: Let α be a positive real number. Let f : ℝ → ℝ and g : α , ∞ → ℝ be the functions defined by f x = sin π x 12 and g x =

lim x → α + g x = lim x → α + 2 log e x - α log e e x - e α = lim x → α + 2 x - α 1 2 x 1 e x - e α 1 2 x e x = lim x → α + e x - e α x - α 1 e x · 2 = 2 e α lim x → α + e x · 1 2 x 1 2 x = 2 lim x → α + f g x = f lim x → α + g x = f 2 = sin π 6 = 1 2 = 0 . 50

Q: The value of the limit lim x → π 2 4 2 sin 3 x + sin x 2 sin 2 x sin 3 x 2 + cos 5 x 2 - 2 + 2 cos 2 x + cos 3 x 2 is _________

lim x → π 2 8 2 sin 2 x · cos x cos x 2 - cos 7 x 2 + cos 5 x 2 - 2 · 2 cos 2 x + cos 3 x 2 = lim x → π 2 8 2 sin 2 x · cos x cos x 2 - cos 3 x 2 + cos 5 x 2 - cos 7 x 2 - 2 2 cos 2 x = lim x → π 2 16 2 sin x cos x · cos x 2 sin x sin x 2 + 2 sin 3 x sin x 2 - 2 2 cos 2 x = lim x → π 2 16 2 sin x cos x · cos x 2 sin x 2 sin x + sin 3 x - 2 2 cos 2 x = lim x → π 2 16 2 sin x cos 2 x 2 sin x 2 2 si

Question 1

Let and be the real numbers such that _ x 0 1 x^3 ( 2 _0^x 1 1-t^2 d t+ x x )=2. Then the value of + is _______

Accepted Answer

_ x 0 2 _0^x 1 1-t^2 d t+ x x x^3 aligned & = _ x 0 2 ( 1 1-x^2 )+ x- x x 3 x^2 \ & = 2 (1-x^2 )^ -1 + (1- x^2 2! + x^4 4! )- x (x- x^3 3! + x^5 5! ) 3 x^2 \ & = 2 (1+x^2+x^4 )+ (1- x^2 2! + x^4 4! )- (x^2- x^4 3! ) 3 x^2 aligned aligned & = ( 2 + )+ x ^2 ( 2 - 2 - )+ x ^4() 3 x ^2 =2( Given ) \ & 2 + =0 and -3 6 =2 \ & =-2 and =12+3 \ & =- 12 5 and = 24 5 aligned + = 12 5 =2.40

Question 2

Let x_0 be the real number such that e^ x_0 +x_0=0. For a given real number , define g(x)= 3 x e^x+3 x- e^x- x 3 (e^x+1 ) for all real numbers x. Then which one of the following statements is TRUE?

Accepted Answer

aligned & e^ x_0 +x_0=0 \ & g(x)= 3 x e^x+3 x- e^x- x 3 (e^x+1 ) \ & g(x)=x- (e^x+x ) 3 (e^x+1 ) , g(x)+e^ x_0 =x+e^ x_0 - 3 ( e^x+x e^x+1 ) \ & g(x)+e^ x_0 = (x-x_0 )- 3 ( e^x+x e^x+1 ) \ & As very clear g (x_0 )+x_0 0 aligned aligned & _ x x_0 | g(x)+e^ x_0 x-x_0 | 0 0 = _ x x_0 | g^ (x) 1 |= |g^ (x_0 ) | \ & g^ (x)=1- 3 ( (e^x+1 ) (e^x+1 )- (e^x+x ) e^x (e^x+1 )^2 ) \ & g^ (x_0 )=1- 3 ( (e^ x_0 +1 )^2 (e^ x_0 +1 )^2 )=1- 3 \ & _ x x_0 | g(x)+e^ x_0 x-x_0 |= |g^ (x_0 ) |= |1- 3 | aligne

Question 3

Let k R . If _ x 0^+ ( ( k x)+ x+x)^ 2 x =e^6, then the value of k is

Accepted Answer

$ aligned & l= _ x 0^ + ( ( k x)+ x+x)^ 2 x =e^6 \ & l= _ x 0^ + 2 x ( ( k x)+ x+x-1) \ & l= _ x 0^ + 2 ( ( k x) k x k x k x k x x +1- (1- x) x^2 x ) \ & l=2(k+1) \ & l=e^ 2(k+1) =e^6 \ & k+1=3 \ & k=2 aligned $

Question 4

Let S be the set of all ( , ) R R such that _ x (x^2 ) ( _e x )^ ( 1 x^2 ) x^ ( _e(1+x) )^ =0. Then which of the following is (are) correct?

Accepted Answer

aligned & _ x x^2 ( _e x )^ 1 x^2 x^ ( _e(1+x) )^ =0 \ & _ x ( _e x )^ ( _e(x+1) )^ x^ +2 =0 aligned aligned & _ x ( _e x _e(x+1) )^ ( _e x )^ - x^ +2 =0 \ & _ x ( _e x )^ - x^ +2 =0 Put _e x=t \ & _ t t^ - (e^t )^ +2 =0 aligned As we know _ x x e^x =0 +2>0 >-2

Question 5

Let the function f:[1, ) R be defined by f(t)= \ array cc (-1)^ n+1 2, & if t=2 n-1, n N , \ (2 n+1-t) 2 f(2 n-1)+ (t-(2 n-1)) 2 f(2 n+1), & if 2 n-1 < t < 2 n+1, n N . array . Define g(x)= _1^x f(t)

Accepted Answer

f ( t )= \ array ccc 2 & ; & t =1 \ 4-2 t & ; & 1

Question 6

Let f(x) be a continuously differentiable function on the interval (0, ) such that f(1)=2 and t x t^ 10 f(x)-x^ 10 f(t) t^9-x^9 =1 for each x>0. Then, for all x>0, f(x) is equal to

Accepted Answer

aligned & _ t x t^ 10 f(x)-x^ 10 f(t) t^9-x^9 =1 \ & _ t x 10 t^9 f(x)-x^ 10 f^ (t) 9 t^8 =1 aligned aligned & 10 x f(x)-x^2 f^ (x)=9 \ & x^2 f^ (x)=10 x f(x)-9 \ & f^ (x)= 10 f(x) x - 9 x^2 \ & d y d x - 10 x y=- 9 x^2 \ & y 1 x^ 10 = - 9 x^2 1 x^ 10 d x \ & y x^ 10 = 9 11 x^ 11 +c...(1) aligned aligned & f (1)=2 2 1 = 9 11 + c c = 13 11 \ & f ( x )= 9 11 x + 13 11 x ^ 10 aligned Option (2) is correct.

Question 7

Let f : 0 , &#160; 1 &#8594; &#8477; be the function defined as f x = n if x &#8712; [ 1 n + 1 , &#160; 1 n ) where n &#8712; &#8469; . Let g : 0 , &#160; 1 &#8594; &#8477; be a function such that &#8

Accepted Answer

Given, f : 0 , &#160; 1 &#8594; &#8477; be the function defined as f x = n if x &#8712; [ 1 n + 1 , &#160; 1 n ) where n &#8712; &#8469; And  g : 0 , &#160; 1 &#8594; &#8477; be a function such that &#8747; x 2 x 1 - t t d t &#60; g x &#60; 2 x for all x &#8712; 0 , &#160; 1 .  Now we need to solve 1 sided limit here to get some answer, as   lim x &#8594; 0 -  doesn&rsquo;t exist here (not in domain) As  1 n + 1 &#8804; x &#60; &#160; 1 n &#8658; n + 1 &#8805; 1 x &

Question 8

If &#946; = lim x &#8594; 0 e x 3 - 1 - x 3 1 3 + 1 - x 2 1 2 - 1 sin x x sin 2 x , then the value of 6 &#946; is

Accepted Answer

&#946; = lim x &#8594; 0 e x 3 - 1 - x 3 1 3 + 1 - x 2 1 2 - 1 sin x x &#183; sin 2 x x 2 &#183; x 2 &#946; = lim x &#8594; 0 1 + x 3 + x 6 2 ! + &#8943; - 1 - 1 3 x 3 + - 1 9 x 6 + &#8943; + - 1 2 x 2 - 1 8 x 4 + &#8943; x - x 3 3 ! + &#8943; x 3 &#946; = lim x &#8594; 0 x 3 1 + 1 3 - 1 2 x 3   (Neglecting higher powers of x ) &#946; = 5 6   i.e.  6 &#946; = 5

Question 9

Let &#945; be a positive real number. Let f : &#8477; &#8594; &#8477;  and g : &#945; , &#8734; &#8594; &#8477;  be the functions defined by  f x = sin &#960; x 12  and  g x =

Accepted Answer

lim x &#8594; &#945; + g x = lim x &#8594; &#945; + 2 log e x - &#945; log e e x - e &#945; = lim x &#8594; &#945; + 2 x - &#945; 1 2 x 1 e x - e &#945; 1 2 x e x = lim x &#8594; &#945; + e x - e &#945; x - &#945; 1 e x &#183; 2 = 2 e &#945; lim x &#8594; &#945; + e x &#183; 1 2 x 1 2 x = 2 lim x &#8594; &#945; + f g x = f lim x &#8594; &#945; + g x = f 2 = sin &#960; 6 = 1 2 = 0 . 50

Question 10

The value of the limit  lim x &#8594; &#960; 2 4 2 sin 3 x + sin x 2 sin 2 x sin 3 x 2 + cos 5 x 2 - 2 + 2 cos 2 x + cos 3 x 2  is _________

Accepted Answer

lim x &#8594; &#960; 2 8 2 &#160; sin 2 x &#183; cos x cos x 2 - cos 7 x 2 + cos 5 x 2 - 2 &#183; 2 cos 2 x + cos 3 x 2 = lim x &#8594; &#960; 2 8 2 &#160; sin 2 x &#183; cos x cos x 2 - cos 3 x 2 + cos 5 x 2 - cos 7 x 2 - 2 2 cos 2 x = lim x &#8594; &#960; 2 16 2 sin x cos x &#183; cos x 2 sin x sin x 2 + 2 sin 3 x sin x 2 - 2 2 cos 2 x = lim x &#8594; &#960; 2 16 2 sin x cos x &#183; cos x 2 sin x 2 sin x + sin 3 x - 2 2 cos 2 x = lim x &#8594; &#960; 2 16 2 sin x &#160; cos 2 x 2 sin x 2 2 si

Question 11

Let e denote the base of the natural logarithm. The value of the real number a for which the right-hand limit lim x &#8594; 0 + 1 &#8722; x 1 &#967; &#8722; e &#8722; 1 x a is equal to a non-zero real

Accepted Answer

lim x &#8594; 0 + 1 &#8722; x 1 x a x a &#8722; 1 d d x ln 1 &#8722; x x = lim x &#8594; 0 + 1 &#8722; x 1 x a x a &#8722; 1 1 x x &#8722; 1 &#8722; ln 1 &#8722; x x 2 = lim x &#8594; 0 + 1 &#8722; x 1 x a x a &#8722; 1 x &#8722; x &#8722; 1 ln 1 &#8722; x x 2 x &#8722; 1 = lim x &#8594; 0 + 1 &#8722; x 1 x x + x &#8722; 1 x + x 2 2 + x 3 3 . . . . . . . . &#8722; a 1 &#8722; x x a + 1 = lim x &#8594; 0 + 1 &#8722; x 1 x x 2 2 + x 3 1 2 &#8722; 1 3 + x 4 1 3 &#8722; 1 4 + &#8230; &#8722; a 1 &#8

Question 12

Let F : R &#8594; R be a function. We say that f has:   PROPERTY 1   if lim h &#8594; 0 &#8289; f h - f 0 h exists and is finite, and  PROPERTY 2   if lim h &#8594; 0 &#8289; f h -

Accepted Answer

A f x = x lim h → 0 ⁡ h - 0 h = lim h → 0 ⁡ h = 0 B f x = x 2 3 lim h → 0 ⁡ h 2 3 - 0 h = lim h → 0 ⁡ h 1 6 = 0 C f x = x x lim h → 0 ⁡ h h - 0 h 2 = lim h → 0 ⁡ h h = 1 , i f h → 0 + - 1 , i f h → 0 - Does not exist D f x = sin ⁡ x lim h → 0 ⁡ sin ⁡ h - 0 h 2 = lim h → 0 ⁡ 1 h sin ⁡ h h ⇒ does not exist

Question 13

For a &#8712; R , &#160; a &#62; 1 , let lim n &#8594; &#8734; &#8289; 1 + 2 3 + &#8230; + n 3 n 7 / 3 1 a n + 1 2 + 1 a n + 2 2 + &#8230; + 1 a n + n 2 = 54 . Then the possible value(s) of a is/are:

Accepted Answer

Let S = lim n → ∞ ⁡ 1 + 2 1 3 + 3 1 3 + … + n 1 3 n 7 3 1 a n + 1 2 + … + 1 a n + n 2 ⇒ S = lim n → ∞ ⁡ ∑ r = 1 n r 1 3 n 7 3 ∑ r = 1 n 1 n a + r 2 S = lim n → ∞ ⁡ ∑ r = 1 n r n 1 3 1 n × n 1 3 n 7 3 ∑ r = 1 n 1 a + r n 2 1 n 2 S = lim n → ∞ ⁡ ∑ r = 1 n r n 1 3 . 1 n ∑ r = 1 n 1 a + r n 2 1 n S = ∫ 0 1 x 1 3 d x ∫ 0 1 1 a + x 2 . d x S = x 4 3 4 3 0 1 - 1 x + a 0 1 S = 3 4 1 a - 1 a + 1 = 54 1 a a + 1 = 1 72 a 2 + a - 72 = 0 a = 8 or a = - 9

Limits — Previous Year Questions & Solutions