Q: If f(x)= \ array l x^2-1,0 < x < 2 \\ 2 x+3,2 x < 3 array ., then the quadratic equation whose roots are _ x 2^ - f(x) and _ x 2^ + f(x) is

Since, f(x)= cases x^2-1, & 0 Now, _ x 2^ - f(x)= _ x 2^ - (x^2-1 ) aligned & = _ h 0 [(2-h)^2-1 ]= _ h 0 [4+h^2-4 h-1 ] \\ & = _ h 0 [h^2-4 h+3 ]=0-0+3=3 aligned and _ x 2^ + f(x)= _ x 2^ + (2 x+3) = _ h 0 [2(2+h)+3]= _ h 0 [4+2 h+3]=7 Hence the quadratic equation whose roots are 3 and 7 is given by x^2-(3+7) x+(3+7)=0 x^2 -10 x+21=0

Q: The value of _ x 0 1- (1- x) x^4 is

1- (1- x)=2 ^2 ( 1- x 2 ) =2 ^2 ( ^2 x 2 ) _ x 0 1- (1- x) x^4 = _ x 0 2 ^2 ( ^2 x 2 ) x^4 = _ x 0 2 ^2 ( ^2 x 2 ) ( ^2 x 2 )^2 ^4 x 2 ( x 2 )^4 16 = 1 8

Q: The value of _ x 0 (1+x)^ 1 x -e+ 1 2 e x x^2 is

Let y=(1+x)^ 1 / x y= 1 x (1+x)= 1 x [x- x^2 2 + x^3 3 - x^4 4 + ] =1- x 2 + x^2 3 - x^3 4 + aligned y & =e^ (1- x 2 + x^2 3 + . ) =e e^ ( -x 2 + x^2 3 . . ) \\ & =e [1+ (- x 2 + x^2 3 - )+ 1 2 ! (- x 2 + x^2 3 - )^2+ ] aligned y-e+ 1 2 e x=e x^2 [ 1 3 +0(x)+ 1 2 (- 1 2 +0(x) )^2+ . ] [ 0(x) in terms containing x] y-e+ 1 2 e x x^2 =e [ 1 3 + 1 8 ]= 11 24 e

Q: Value of _ x 0 a^ x -1 x is

We have, _ x 0 a^ x -1 x = _ y 0 a^ y -1 y = a , where y = x x 0 y = x 0

Q: _ x 0 2 ^ 2 3 x x^ 2 is equal to

Consider _ x 0 2 ^ 2 3 x x^ 2 array l =2 _ x 0 [ 3 x x ]^ 2 =2 _ x 0 [3 3 x 3 x ]^ 2 \\ =2.9 _ x 0 ( 3 x 3 x )^ 2 =18 1=18 array

Q: _ x 0 x- x x+ ^ 2 x is equal to

_ x 0 x- x x+ ^ 2 x = _ x 0 1- x x 1+ ^ 2 x x = _ x 0 1- x x 1+ ( x x ) x = 1-1 1+1 0 =0

Question 1

Given a real valued function ' f ' such that f(x)= \ array cc ^2\ x\ x^2-[x]^2 & for x 0 \ 1 & for x=0 \ \ x\ \ x\ & for x 0 array . then

Accepted Answer

_ x 0^ - f(x)= _ h 0 \ -h\ \ -h\ = _ h 0 (1-h) (1-h) = 1 _ x 0^ + f(x)= _ h 0 ^2\ h\ h^2-[h]^2 = _ h 0 ^2 h h^2 =1 _ x 0 f(x) does not exist.

Question 2

Let f(x)= x, g(x)= x, h(x)=x^2 then _ x 1 f(g(h(x)))-f(g(h(1))) x-1 =

Accepted Answer

Given f(x)= x, g(x)= x, h(x)=x^2 _ x 1 f(g(h(x)))-f(g(h(1))) x-1 When limit it applied, it gives 0 0 form, so we apply L'Hospital rule. _ x 1 ( x^2 ) (- x^2 )(2 x)-0 1-0 Apply the limit, -2(1) (1) ( 1) =-2 (1) ( 1)

Question 3

If f(x)= \ array l x^2-1,0 < x < 2 \ 2 x+3,2 x < 3 array ., then the quadratic equation whose roots are _ x 2^ - f(x) and _ x 2^ + f(x) is

Accepted Answer

Since, f(x)= cases x^2-1, & 0 Now, _ x 2^ - f(x)= _ x 2^ - (x^2-1 ) aligned & = _ h 0 [(2-h)^2-1 ]= _ h 0 [4+h^2-4 h-1 ] \ & = _ h 0 [h^2-4 h+3 ]=0-0+3=3 aligned and _ x 2^ + f(x)= _ x 2^ + (2 x+3) = _ h 0 [2(2+h)+3]= _ h 0 [4+2 h+3]=7 Hence the quadratic equation whose roots are 3 and 7 is given by x^2-(3+7) x+(3+7)=0 x^2 -10 x+21=0

Question 4

The value of _ x 0 1- (1- x) x^4 is

Accepted Answer

1- (1- x)=2 ^2 ( 1- x 2 ) =2 ^2 ( ^2 x 2 ) _ x 0 1- (1- x) x^4 = _ x 0 2 ^2 ( ^2 x 2 ) x^4 = _ x 0 2 ^2 ( ^2 x 2 ) ( ^2 x 2 )^2 ^4 x 2 ( x 2 )^4 16 = 1 8

Question 5

The value of _ x 0 (1+x)^ 1 x -e+ 1 2 e x x^2 is

Accepted Answer

Let y=(1+x)^ 1 / x y= 1 x (1+x)= 1 x [x- x^2 2 + x^3 3 - x^4 4 + ] =1- x 2 + x^2 3 - x^3 4 + aligned y & =e^ (1- x 2 + x^2 3 + . ) =e e^ ( -x 2 + x^2 3 . . ) \ & =e [1+ (- x 2 + x^2 3 - )+ 1 2 ! (- x 2 + x^2 3 - )^2+ ] aligned y-e+ 1 2 e x=e x^2 [ 1 3 +0(x)+ 1 2 (- 1 2 +0(x) )^2+ . ] [ 0(x) in terms containing x] y-e+ 1 2 e x x^2 =e [ 1 3 + 1 8 ]= 11 24 e

Question 6

Value of _ x 0 a^ x -1 x is

Accepted Answer

We have, _ x 0 a^ x -1 x = _ y 0 a^ y -1 y = a , where y = x x 0 y = x 0

Question 7

_ x 0 2 ^ 2 3 x x^ 2 is equal to

Accepted Answer

Consider _ x 0 2 ^ 2 3 x x^ 2 array l =2 _ x 0 [ 3 x x ]^ 2 =2 _ x 0 [3 3 x 3 x ]^ 2 \ =2.9 _ x 0 ( 3 x 3 x )^ 2 =18 1=18 array

Question 8

_ x 0 x- x x+ ^ 2 x is equal to

Accepted Answer

_ x 0 x- x x+ ^ 2 x = _ x 0 1- x x 1+ ^ 2 x x = _ x 0 1- x x 1+ ( x x ) x = 1-1 1+1 0 =0

Limits — Previous Year Questions & Solutions