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

(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))

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

Match the following : {:("Column I ", " Column II" ), ("(A)" if lim_(x to 1) (1-x) tan""(pix)/2 = k " then " sin (1/k) " is" , "(p)"4),( "(B)" if lim_(x to 5) (x^(k)-5^(k))/(x-5) = 500 " then k is " , "(q)" 1),("(C)" lim_(x to oo)(1 + 4/(x+1))^((3x-1)/3) " is equal to " e^(k) " , then k is" , "(r) A perfect sqare"), ("(D) " d^(20)/(dx^(20)) (2 cos x"," cos 3x)= 2^(4k) [ cos2x + 2^(20) . cos4k]" then k is " , "(s)" 5),(, "(t)An odd number"):}

Match the following : {:("Column I ", " Column II" ), ("(A)" if lim_(x to 1) (1-x) tan""(pix)/2 = k " then " sin (1/k) " is" , "(p)"4),( "(B)" if lim_(x to 5) (x^(k)-5^(k))/(x-5) = 500 " then k is " , "(q)" 1),("(C)" lim_(x to oo)(1 + 4/(x+1))^((3x-1)/3) " is equal to " e^(k) " , then k is" , "(r) A perfect sqare"), ("(D) " d^(20)/(dx^(20)) (2 cos x"," cos 3x)= 2^(4k) [ cos2x + 2^(20) . cos4k]" then k is " , "(s)" 5),(, "(t)An odd number"):}

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

Prove that underset(xrarra)"lim"((x+5)^(5//2)-(a+5)^(5//2))/(x-a)=(5)/(2)(a+5)^(3//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]]