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

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

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

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

If f(x) is differentiable and strictly increasing function,then the value of lim_(x rarr0)(f(x^(2))-f(x))/(f(x)-f(0)) is 1 (b) 0(c)-1 (d) 2

D*f(x)=lim_(h rarr0)(f^(2)(x+h)-f^(2)(x))/(h) If f(x)=x ln x then D*f(x) at x=e equals

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

If f(x)=|x-2| ,then (a) lim_(x rarr2+)f(x)!=0 (b) lim_(x rarr2-)f(x)!=0 (c) lim_(x rarr2+)f(x)!=lim_(x rarr2-)f(x) (d) f(x) is continuous at x=2

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]]

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=