A question about derivatives in a particular calculation

Question

I am attaching a photo of a calculation from these notes below:

How is $\lim\limits_{r\to 0^+}\frac{\tilde{F}(r+t;x)-\tilde{F}(r-t;x)}{2r}=\partial_t\tilde{F}(t;x)?$

Answer 1

I think there's a typo in the formula: $r-t$ should be $t-r$.

Then it is essentially the definition of the (partial) derivative, written as the limit of the difference quotient (slope of secant line) using for the numerator
$$ F(t+r) - F(t-r) = ( F(t + r) -F(t)) + (F(t) - F(t-r)). $$ When you divide by $r$ and take the limit as $r \to 0$ you get twice the derivative.