A “Paradox” in the Cooling Problem According to Newton's law of cooling, the rate at which an object cools is directly proportional to the difference in temperature between the object and the surrounding medium. Thus, let the temperature of the object be $T$, the temperature of the surrounding medium be a constant $Q(T>Q)$, the proportionality coefficient be $\lambda>0$, and the time be $t$. Then we obtain the differential equation
$$\frac{{\rm d} T}{{\rm d} t}=-\lambda (T-Q).$$
We can readily solve it and derive the general solutions
$$T=Q+Ce^{-\lambda t},$$ where $C$ is any a constant. Assume that $T(0)=T_0$. Then 
the particular solution is that $$T=Q+(T_0-Q)e^{-\lambda t}.$$
Now, imagine that we want to know how long the object takes to cool completely, namely we want to find the solution for $T=Q$, which is
$$(T_0-Q)e^{-\lambda t}=0.$$
This demands that $t \to +\infty$. Therefore, it need take a infinitely long time. But, our experience form the empirical world tells that the time needed is always finite, and even not so long. This seems render a paradox resembling the one "Achilles could never catch the tortoise". How to understand it?
 A: Our experience from the empirical world tells us that:


*

*when the difference of temperature (or of any two physical quantities, for that matter) is smaller than the precision of our instrument, we can't tell them apart.

*"temperature" is a macroscopical concept, which measures the average kynetic energy of a large quantity of particles which move/vibrate, in principle and in point of fact, at different paces.
As per the rate of convergence, keep in mind that $e^{-\lambda t}$ decays quite fast, compared to the parameters of the problem. For instance, when $t$ is just five times the "system time" $\tau=\frac1\lambda$, you'll notice that $e^{-\lambda t}$ is already $\approx 0.007$, which is enough to annihilate  all real-life differences of temperature.
That being said, I wouldn't call real-life cooling or heating a fast process. It takes less time to fall from an airplane than it takes for my soup to cool down to room temperature.
A: As LutzL and SaucyO'Path have pointed out, the imprecision in temperature measurements ensures $T$ is indiscernible from $Q$ within a finite time. I'll just add a more general discussion of when ODEs can reach some steady state in finite time. Consider the equation $$\frac{dx}{dt}=x^q.$$This has solution $$x^{1-q}=x_0^{1-q}+(1-q)t.$$If $q>1$, at $t=\frac{1}{q-1}x_0^{1-q}$ we get $x^{1-q}=0$ or equivalently $x=\infty$. This is an example of what is called a finite-time singularity, and they're encountered in one possible future for the universe and certain heterodox models of stock market crashes.
If $q\le 1$, of course, there's no finite time singularity. For example, $q=1$ yields $x\to\infty$ and $y:=x^{-1}\to 0$, but $y$ (which satisfies $\dot{y}=-y$) doesn't become $0$ in finite time. Most ODEs in mechanics don't reach their $t=\infty$ behaviour in finite time; most PDEs in field theory don't reach their at-infinity behaviour at any finite values for $t,\,\mathbf{x}$. But do they get arbitrarily close to it? Yes, of course; by definition, that's how limits work. Sometimes you can argue a limit is $0$ from normalization alone. (For example, in quantum mechanics some $\psi$ satisfies a differential equation, with $|\psi|^2$ a PDF, so $\psi\to 0$ as $x\to\infty$.)
