$\lim_{s\to1}\frac{t-s}{1-s}$ when $0\leq s\leq t\leq 1$

Question

I was working on a topology exercise and got to a point (no pun intended) where I had to analyze the behaviour of $f(t,s)=\frac{t-s}{1-s}$ as s goes to $1$, under the assumption that $t\geq s$. It seems pretty obvious that the limit should exist and be equal to $1$, but I haven't been able to prove this. I'm still thinking about it, but in the meantime I figured I would ask here for advice on how to approach this. Of course I haven't ruled out the possibility that the solution is right in front of me, but I don't see it at the moment.

EDIT: I forgot to mention that $0\leq t,s\leq 1$.

Answer 1

Let $t=s+\lambda(1-s)$ with $0\le \lambda\le 1$ then

$$\frac{t-s}{1-s}=\frac{\lambda(1-s)}{1-s}=\lambda$$

therefore the limit doesn't exist.

Answer 2

The limit does not exist unless $t-s$ tends to $0$. If $t$ does not depend on $s$ then $t$ must be equal to $1$ for the limit to exist.

If $t$ does depend on $s$ then anything can happen. Try for example $t=2s$ and $t=s$.