kinematics - Is $dfrac{dx}{dt}$ a fraction or not?

I am new to calculus and during my mathematics class my sir defined $\dfrac{dx}{dt}$ as

$$dx/dt=\lim_{t\to t_1}\dfrac{f(t)-f(t_1)}{t-t_1}$$ and my sir made a clear statement that

$\dfrac{dx}{dt}$ is not a fraction it only behaves like a fraction!

(it means $\dfrac{dx}{dt}$ is just a notation to represent that big limit!) and he made a statement that

$dx$ or $dt$ does not have any meaning it is just $\dfrac{d}{dt}(x)$ which has meaning but we treat it as $\dfrac{dx}{dt}$.

but at same time my physics sir, to derive velocity he stated

let the particle be at position $x$ at time $t$ and after an infinitesimal change in position and time it reaches $x+dx$ at time $t+dt$. Now velocity is $\dfrac{displacement}{time}$ , so we will get $v=\dfrac{dx}{dt}$

and this expression purely tells that $\dfrac{dx}{dt}$ is a fraction!

Now i don't know who is correct, so please help!