(37)quad (d)/(dx)[lim_(x rarr a)(x^(5)-a^(5))/(x-a)]=

lim_(x rarr0)(a^(x)-b^(x))/(c^(x)-d^(x))

lim_(x rarr0)(a^(x)+b^(x)-c^(x)-d^(x))/(x)

Which of the following is/are true? (a) lim_(x rarr oo)((2+x)^(40)(4+x)^(5))/((2-x)^(45))=1(b)lim_(x rarr0)(1-cos^(3)x)/(x sin x cos x)=(3)/(2)(c)lim_(x rarr0)(ln(1+2x)-2ln(1+x))/(cot^(-1)(sqrt(x+1)-sqrt(x)))=-1 (d) lim_(x rarr oo)(cot^(-1)(sqrt(x+1)-sqrt(x)))/(sec^(1)((2x+1)/((x-1)^(2)))=1)=-1 (d)

let a=lim_(x rarr1)((x)/(ln x)-(1)/(x ln x)),b=lim_(x rarr0)((x^(3)-16x)/(4x+x^(2))),c=lim_(x rarr0)(ln(1+sin x))/(x) and d=lim_(x rarr-1)((x+1)^(3))/(3[sin(x+1)-(x+1)]) then the matrix [[a,bc,d]]

If p= lim_(x rarr0^(+))[(3sin x)/(x)] ,q= lim_(x rarr0^(+))[(3x)/(sin x)] ,r= lim_(x rarr0^(+))[(tan x)/(x)] and s=lim_(x rarr0^(+))[(3tan x)/(x)] where [.] denotes the greatest integer function (A) pqrs=18 (B) pqrs=0 (C) p+q+r+s=9 (D) pq+rs+qr+sp=18

Let f be a positive differentiable function defined on (0,oo) and phi(x)=lim_(nrarroo) (f(x+(1)/(n))/f(x))^(n) . Then intlog_(e)(phi(x))dx=

A differential equation of the form dy/dx+Py=Q is said to be a linear differential equation. Integrating factor of this differential equation is e^int Pdx and its solution is given by y.e^(int Pdx)=int (Qe^(int Pdx))dx+c . Answer the question:Let f(x) be a differentiable function in intervel (0, oo) such that f(1)=1 and lim_(trarrx) (t^2f(x)-x^2f(t))/(t-x)=1 for all x gt 0 . Then f(x) = (A) 1/(3x)+(2x^2)/3 (B) -1/(3x)+(4x^2)/3 (C) -1/x+2/x^2 (D) 1/x

Which of the following true ({.} denotes the fractional part of the function)? lim_(x rarr oo)(log_(e)x)/({x})=oo( b) lim_(x rarr2^(+))(x)/(x^(2)-x-2)=oolim_(x rarr1^(-))(x)/(x^(2)-x-2)=-oo( d) lim_(x rarr oo)(log_(0)5x)/({x})=oo

Which of the following limits does not exist? (a) lim_(x rarr oo)cos ec^(-1)((x)/(x+7)(B)lim_(x rarr1)sec^(-1)(sin^(-1)x)(C)lim_(x rarr0^(+))x^((1)/(x))(D)lim_(x rarr0)(tan((pi)/(8)+x))^(cot x)

If G(x)=-sqrt(25-x^(2)), then lim_(x rarr1)(G(x)-G(1))/(x-1)is (a) (1)/(24) (b) (1)/(5)(c)-sqrt(24) (d) none of these