Evaluate `lim_((x,y)->(0,0))(1+x^2)(siny)/y` `y ne 0`

Evaluate: lim_(x rarr0)(y^(2)+sin x)/(x^(2)+sin y^(2)) where (x,y)rarr(0,) along the curve x=y^(2)

Evaluate : lim_(x to 0)f(x)={{:(abs(x)/x",",x ne 0), (" 0,", x=0.):}

lim_(x rarr 0) x^2/y = 0

Find the general solution of the differential equation (x + 2y^(3)) (dy)/(dx) =y, y ne 0

Q.Evaluate lim_(y rarr0)[lm it_(x rarr0)(exp(x ln(1+a(y)/(x)))-exp(x ln(1+b(y)/(x))))/(y)

Let f(x,y)=sin(2x-y)cos y+cos(2x-y)sin y for all x,y in R. The value of lim_(x rarr0)(1+tan x-x)^((1)/(x^(2)f(x,y))) is equal to

Solve the following differential equations. ( 2x -10 y^2) dy + y dx =0 , y ne 0

(i) lim_(x to 0) (x tan 4x)/(1-cos 4x) (ii) lim_(y to 0) ((x+y)sec(x+y)-x sec x)/(y )

If (x_(0), y_(0), z_(0)) is any solution of the system of equations 2x-y-z=1, -x-y+2z=1 and x-2y+z=2 , then the value of (x_(0)^(2)-y_(0)^(2)+1)/(z_(0)) (where, z_(0) ne 0 ) is