For 0 < x < y, show that 1/ y < (ln (y) - ln (x) )/ (y - x) < 1/ x

Question

I believe this might have something to do with intergration but not sure please help

Answer 1

Since 0<x<y, y>x, y/x>1. ln(y)-ln(x)=ln(y/x)>0; y-x=x(y/x-1).

(ln(y)-ln(x))/(y-x)=ln(y/x)/(y-x).

Let y/x=1+p, so ln(y/x)=ln(1+p)=p-p²/2+p³/3-...;

Consider x→y, then p is small, so ln(y/x)=ln(1+p)<p, that is, <y/x-1, <(y-x)/x.

So (ln(y)-ln(x))/(y-x)→[(y-x)/x]/(y-x)<1/x.

Consider x=y/n, where n>1, then y/x=n,

so (ln(y)-ln(x))/(y-x)=ln(n)/(nx-x)=ln(n)/[(y/n)(n-1))]=nln(n)/(y(n-1)).

When n is very large n/(n-1)→1 and x→0 so (ln(y)-ln(x))/(y-x)→ln(n)/y>1/y because ln(n)>>1 when is large. Therefore, 

1/y<(ln(y)-ln(x))/(y-x)<1/x.