TITLE: What does "approach zero" really mean?
QUESTION [33 upvotes]: We repeat the phrase "approach zero" regularly, but what exactly does it mean to say "$\delta t$ approaches zero"? 
P.S. If this a stupid question, then please forgive me.

REPLY [24 votes]: Your question isn't stupid, it's the heart of calculus.
An introductory step from algebra to calculus is in the context of slope.  Algebra allows us to find an average slope, while calculus allows us to find the instantaneous slope.  In other words, algebra gives us the slope of a line, while calculus gives us the slope of a point.
Slope of a point?  Yes, but let's stay with lines for now.  The slope of a line is given by the following function.  We divide the change in $y$ by the change in $x$:
$$m=\frac{y_2 - y_1}{x_2 - x_1}$$
However, if the line is curved, say $f(x)=x^2$, then each point on that line has a different slope.  If we are asked to find the slope when $x=5$, then we can approximate it by finding the average slope between $x=4$ and $x=6$:
$$m=\frac{6^2-4^2}{6-4}$$
But this isn't the correct answer.  If we wanted to get closer to the correct answer, we would choose values that are closer to 5:
$$m=\frac{5.1^2-4.9^2}{5.1-4.9}$$
The pattern to notice is that the more accurate our answer becomes, the smaller the difference between $x_2$ and $x_1$.  In fact, the correct answer will be found when the difference is zero.  However, when we go to write this down, we have a problem:
$$m=\frac{0}{0}$$
Specifically, we can't divide by zero.  We've gone as far as algebra can take us, and we need a new way to talk about math.  We need calculus.  In algebra, we saw that we get closer and closer to the correct answer.  In calculus, this is called the "limit".  We get closer and closer to the limit as the divisor gets closer and closer to zero.  The divisor "approaches zero".
Finally, we have "What is the limit as x approaches zero?"