If `x_1 and x_2` are the real and distinct roots of `a x^2+b x+c=0,` then prove that `lim_(n->x_1){1+"sin"(a x^2+b x+c)}^(1/(x-x_1))=e^(a(x_1-x_2))`

If x_(1) and x_(2) are the real and distinct roots of ax^(2)+bx+c=0 then prove that lim_(xtox1) (1+sin(ax^(2)+bx+c))^((1)/(x-x_(1)))=e^(a(x_(1)-x_(2))).

If x_1 and x_2 are two distinct roots of the equation acosx+bsinx=c , then tan((x_1+x_2)/2) is equal to (a) a/b (b) b/a (c) c/a (d) a/c

The number of distinct real roots of x^4-4x^3+12 x ^2+x-1=0 is ________

Evaluate: lim_(x->2)(sin(e^(x-2)-1))/(log(x-1))

Evaluate lim_(x to 0) x^2 sin (1/x) .

If f(x)=sin^(-1)x then prove that lim_(x->1/2)f(3x-4x^3)=pi-3lim_(x->1/2)sin^(-1)x

If alpha,beta are the roots of the equation a x^2+b x+c=0, then find the roots of the equation a x^2-b x(x-1)+c(x-1)^2=0 in term of alphaa n dbetadot

Evaluate lim_(xto1) (x^(2)+xlog_(e)x-log_(e)x-1)/((x^(2))-1)

The value of lim_(xrarr0) ((1+2x)/(1+3x))^((1)/(x^(2))).e^((1)/(e^(x))) is

lim_(x->0) ((1+x)^(1/x)-e +(ex)/2)/(x^2)