Definition of Derivative and the Delta

Question

I am doing some approximation for a function, during this task I came across the following equation and I was wondering if I can consider it as the derivative with respect to the variable $t$: $$ \lim\limits_{h\to 0} \frac{f(x,t+ht)-f(x,t)}{h^2t} $$ .

Thanks in advance.

Answer 1

No, simple counterexample:

Let $f(x, t) = t$. Then

\begin{align*} \frac{f(x, t+ht) - f(x, t)}{h^2 t} &= \frac{t+ht-t}{h^2 t} = \frac{1}{h} \overset{h \to 0}{\longrightarrow} \infty, \end{align*}

but $\frac{\partial}{\partial t}f(x,t) = 1.$

Answer 2

Nearly. If $t\ne0$,

$$\lim\limits_{h\to 0} \frac{f(x,t+ht)-f(x,t)}{ht}=\dfrac{\partial f(x,t)}{\partial t}$$

and if $t=0$, the limit is $0$.