solve x/x^ 2-1 by definition of derivative

Question

I NEED TO SOLVE THIS PROBLEM BY USING THE DEFINITION OF LIMITS.

Answer 1

Let y=x/(x²-1).

Let h be a small change in x which creates a small change k in y:

y+k=(x+h)/((x+h)²-1),

So k=(x+h)/((x+h)²-1)-y=(x+h)/((x+h)²-1)-x/(x²-1).

k=((x+h)(x²-1)-x((x+h)²-1))/(((x+h)²-1)(x²-1)).

(x+h)²=x²+2xh+h²≈x²+2xh when h is small.

k=(x³-x+hx²-h-x(x²+2xh-1)/((x²+2xh-1)(x²-1)),

k=(x³-x+hx²-h-x³-2x²h+x)/((x²-1+2xh)(x²-1)),

k=-h(x²+1)/((x²-1)²+2xh(x²-1)),

dy/dx=k/h=-(x²+1)/(x²-1)² as h→0.

Therefore the derivative is -(x²+1)/(x²-1)².