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**vanishingpoint** Hi, everyone! I have a minor problem proving the two following definitions of the derivative are equivalent:

f is differentiable iff $\displaystyle \lim_{x->x_{0}}\frac{f(x)-f(x_{0})}{x-x_{0}}$ exists

and

f is differentiable iff $\displaystyle \lim_{t->0}\frac{f(x_{0}+t)-f(x_{0})}{t}$ exists.

I know that i can let $\displaystyle t=x - x_{0}$, so that $\displaystyle x=x_{0}+t $ and $\displaystyle \frac{f(x) - f(x_{0})}{x - x_{0}}=\frac{f(x_{0} + t) - f(x_{0})}{t}$, but then how can i explain the limits part, when in first case i am taking limit as x approaches $\displaystyle x_{0}$ and in second as t approaches 0?

Thanks in advance!