Differentiable, continuous function... "show that"

If $\displaystyle f$ is continuous on $\displaystyle [0,1]$ and differentiable at a point $\displaystyle x \in [0,1]$, show that, for some pair $\displaystyle m,n \in \mathbb{N}$,

$\displaystyle \left | \frac{f(t)-f(x)}{t-x}\right | \leq n$ whenever $\displaystyle 0 \leq |t-x| \leq \frac{1}{m}$

Since it's differentiable at x I know $\displaystyle \lim_{t \to x}\frac{f(t)-f(x)}{t-x}$ exists...

And it's continuous so I know for that for any $\displaystyle \epsilon > 0$ there's a $\displaystyle \delta > 0$ so that $\displaystyle |f(t)-f(x)| < \epsilon$ when $\displaystyle |t-x| < \delta$

Not sure how to put it all together...