Limits: JEE Questions & Solutions

Let f: R (0,) be a twice differentiable function such that f(3) = 18, f'(3)=0 and f''(3) = 4. Then x 1 e [ f(2+x)f(3) ]^ 18(x-1)^2 is equal…

If x 0 e^(a-1)x+2 bx+(c-2)e^-x x x- e(1+x) =2, then a^2+b^2+c^2 is equal to

If the function f(x)= e^x (e^ x - x-1)+ e( x+ x)-x x-x is continuous at x=0, then the value of f(0) is equal to

The value of x 0 e ( (ex) (e^2x) (e^10x)) e^2-e^2 x is equal to:

If x ( e1 - e ( 1e - x1 + x))^x =, then the value of e 1 + e equals:

Evaluate the following limit: x (2x^2 - 3x + 5) ( 3x - 1)^x/2(3x^2 + 5x + 4) (3x + 2)^x. The value of the limit is:

Evaluate the limit: x 0 x ( 2 ^2x + 3 x - ^2x + x + 4) is equal to:

Let f(x) = n r=0^n ( ( x2^r+1) + ^3 ( x2^r+1)1 - ^2 ( x2^r+1)) Then, x 0 e^x - e^f(x)x - f(x) is equal to:

Let [t] be the greatest integer less than or equal to t. Then the least value of p N for which x ( x ( [ 1/x ] + [ 2/x ] + + [ p/x ]) - x^2…

If t ( 0^1 ( 3x + 5)^t dx) = 5e ( 85)^ 32, then is equal to:

For,, R, if x 0 x^2 x + ( - 1)e^x^2 - 3 2x - x = 3, then + - is equal to:

If x 0 (2x) + a (4x) - bx^4 is finite, then (a + b) is equal to:

If x 0 ( xx)^ 1x^2 = p, then 96 e p is equal to

If x 1 (x-1)(6+ (x-1)) + (1-x)(x-1)^3 = -1, where, R, then + is equal to

Evaluate the following limit: x 0^+ (5x^ 13) (1 + 3x^2) ( ^-1 (3 x))^2 (e^5x^ 43 - 1)

If the function f(x) = ( x) - ( x) x - x is continuous at x = 0, then f(0) is equal to:

For t -1, let t and t be the roots of the equation ( (t + 2)^ 17 - 1)x^2 + ( (t + 2)^ 16 - 1)x + ( (t + 2)^ 121 - 1) = 0. If t 1^+ t = a…

Given below are two statements: Statement I: x 0 ( ^-1 x + e 1+x1-x - 2xx^5) = 25 Statement II: x 1 ( 2x^1-x) = 1e^2 In the light of the…

If a = x 0 [ (1 + (1 + x^4)) - 2x^4 ] and b = x 0 [ ^2(x) 2 - 1 + x ], then ab^3 is:

If x 0 3 + (x) + (x) + (1 - x)3 ^2(x) = 13, then 2 - is equal to:

Evaluate the limit x /2 [ ( 1x - 2) 0^ /2 ( 1t^3) dt ], which is equal to:

Let the slope of the line 45x + 5y + 3 = 0 be 27r1 + 9r2^2 for some r1, r2 R. Then: x 3 x^3 ( 8t^23r2x^2 - r2x^2 - r1x^3 - 3x)dt is equal…

Let f: - /2, /2 R be a differentiable function such that f(0) = 1/2. If the limit x 0 0^x f(t) dte^x^2 - 1 =, then 8 ^2 is equal to:

Evaluate the limit: x 0 [ e^2| (x)| - 2| (x)| - 1x^2 ]

If x ( f(7x)f(x)) = 1, then x [ ( f(5x)f(x)) - 1 ] is:

Let f: (- 2, 2) R be a differentiable function such that f(0) = 12. If the limit x 0 0^x f(t) dte^x^2 - 1 =, then 8 ^2 is equal to:

If the function f(x) = 72x - 9x - 8x + 1 2 - 1 + (x), x 0; a (e^2) (e^3), x = 0 is continuous at x = 0, then the value of a^2 is equal to:

Let f(x) = 0^x (t + (1 - e^t)) dt, x R. Find x 0 f(x)x^3:

Let f: (- /2, /2) → R be a differentiable function such that f(0) = 1/2. If the limit x 0 0^x f(t) dte^x^2 - 1 =, then 8 ^2 is equal to:

If n (1^2-1)(n-1)+(2^2-2)(n-2)+ +((n-1)^2-(n-1))(1^3+2^3+ +n^3)-(1^2+2^2+ +n^2) is equal to:

The value of n 1 + 2 - 3 + 4 + 5 - 6 + + (3n - 2) + (3n - 1) - 3n 2n^4 + 4n + 3 - n^4 + 5n + 4 is:

Let x = 2 be a root of the equation x^2 + px + q = 0 and f(x) = cases 1 - (x^2 - 4px + q - 8q^2 + 16)(x - 2p)^2, & x 2p, 0, & x = 2p. cases…

x 0 48x^4 0^x t^3t^6 + 1 dt is equal to.

n ( 3n [ 4 + ( 2 + 1n)^2 + ( 2 + 2n)^2 + + ( 3 - 1n)^2 ])

The number of real roots of the equation x^2 - 4x + 3 + x^2 - 9 = 4x^2 - 14x + 6 is:

Evaluate the limit: x 1 ( 3x+1 + 3x-1)^6(x + x^2 - 1)^3 + ( 3x+1 - 3x-1)^6

If n ( 11+n + 12+n + 13+n + + 12n) is equal to:

n ( 12^2 - 2^3) ( 12^2 - 2^5) ( 12^2 - 2^2n+1) is equal to:

x 0 ( (1 - ^2(3x)) ^3(4x) ^3(4x) (2x + 15)) is equal to:

The number of integral solutions of 2 ( x - 72x - 3) 0 is:

Among (S1) n 1n ( 2 + 4 + 6 + + 2n) = 1 (S2) n 116 ( 1^15 + 2^15 + 3^15 + + n^15) = 116

If x 0 e^x - (bx) - cx1 - (2x) = 2, then 5a^2 + b^2 is equal to: