How does instantaneous speed work for circular motion? Why do we use the formula $\lim_{\delta t→0} \delta s/\delta t$ to get the instantaneous speed? Since speed is distance divided by time, what does the limit have to do with this? I have a very limited knowledge regarding calculus. Please answer without calculus if possible. 
 A: Speed is distance divided by the time taken to cover that distance.
Ok now what is instantaneous speed ? Its the speed of the body in consideration at this very instant in time.
How can we measure the speed at this very instant? To measure it, lets take a small time interval $dt$ and say that the object cover $ds$ distance in this interval now speed is simply $ds/dt$ now this speed is not actually the speed at this instant as we are talking about an extended interval of time, $dt$ Only if this interval tends to zero can we get the actual instantaneous speed hence we take limit dt tending to zero.
A: 
Why do we need the concept of limits?  

Calculus is an unavoidable tool in physics. Let's see what do we mean by limits. The limit of a function, $f(x)$, as its variable, $x$, approaches some value, say $a$,  is the value to which the function approaches when $x$ approaches $a$. Let's denote the the approach of $x$ to $a$ as $x\to a$ (spell it as $x$ tends to the value $a$). That is, the fundamental problem in limits is to find the value of $f(x)$ as  $x\to a$. Let's call this function value $f(x\to a)$. This is mathematically stated as follows:  
$$\lim_{x\to a}f(x)=f(x\to a)$$  
In our problem, for a moving particle, its position varies with time and so the displacement is a function of time. So we denote the function as $s(t)$. This notation means the displacement of the particle at an instant $t$ (or say, the position of the particle at $t=t$ with reference to the initial position at $t=t_0$), not in an interval $(t-t_0)$.
A physical example is that consider a moving car. You want to travel from $A$ to $B$. The speed of the car, is of course, not uniform (this is the important thing that why we need calculus). The car accelerates, decelerates and may move with uniform velocity and so on. You can then check the average speed with which you drove the car by taking the ratio of the distance travelled to the time taken for the journey.  However, this is just an average speed. It only tells you that, if you have travelled in somewhat that constant speed, you will cover such a distance in that time. But, you know that's not the velocity of the car at each instant. The speed has been changing, may be you may have stopped your car.
The concept of average holds good (i.e., it gives you a good approximation of sme value) only if the values for which you took the average are all comparable. In our case if the car is moving around some value of $40 \text{ kmph}$, all the time you will get an average around $40 \text{ kmph}$, which can be taken as your speed at any instant to a very good approximation. But, if the car has been stopped for some time, the velocity is zero, which is a very large deviation from the average value and you will never get an accurate result. These are fundamental and you may know that. So, in general, the concept of average velocity does not give any information about the speed of the particle at any instant; it would give under the condition of uniform motion (which is not what we face in reality).  
You may have observed the speedometer of your vehicle. It keeps changing whenever the speed of the vehicle changes. This instrument gives you the velocity at which you are travelling at the present instant. In physics, we need to find out the velocity of the particle at all instants to trace out its trajectory. You see, the purpose of mechanics is to find out the how the particle changes its position with respect to time. That's the solution for every mechanical problems.  

How the concept of limits comes to the rescue?  

We have seen the definition of limits and a real life problem where the average velocity fails. Now, let's see how limits could solve the problem. Actually, limits could give us a hint, but will not exactly solve the problem. Before getting into your problem, let's see an example. Consider the function   
$$f(x)=\frac{1}{\sin x}, \space \space\space 0\le x\le 2\pi$$  
We know that this function has no well defined value at the points $x=0$ and $x=2\pi$, since the function blows up to infinity at these points. Even with limits, we cannot see what is the value of the function at the points $x=0$ and $x=2\pi$. What the limits could tell us is about the value to which the function $f(x)$ approaches as $x$ approaches zero. Since the sine function is symmetrical about these two points, we can analyse any one of these, say $x=0$. We make a table as shown below:  
$$
\begin{array}{|c|c|}
\hline
x& f(x)\\ \hline
1& 1.188395106\\
0.1& 10.01668613\\
0.01& 100.0016667\\
0.001& 1000.000167\\
\vdots& \vdots\\
0.000000001& 1000000000\\
\vdots& \vdots\\ \hline
\end{array}
$$   
You can see that as $x$ approaches the value $0$, the function is increasing so rapidly. An interesting thing we note here is that as $x$ is very close to zero, the function value is approximately $1/x$ (see the last entry).  Hence, we can say that the function changes from $\frac{1}{sin x}$ to $\frac{1}{x}$ as $x\to 0$. That is  
$$\lim_{x\to 0}\frac{1}{sin x}=\frac{1}{x}$$  
This is simply because $\large{sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots\approx x}$, for $x<<1$.  
Now consider another example  
$$g(x)=\frac{sin x}{x}$$  
Of course, there is a singularity at $x=0$, where we get the function value in a $0/0$ form. As we done earlier, we make the table with $x$ values and the corresponding function values:    
$$
\begin{array}{|c|c|}
\hline
x& g(x)\\ \hline
1& 0.841470984\\
0.1& 0.998334166\\
0.01& 0.999983333\\
0.001& 0.9999999833\\
0.0001& 0.999999998\\ 
\vdots& \vdots\\ \hline
\end{array}
$$  
As you can see, the function value approaches $1$ s $x$ approaches $0$. Hence we can say  
$$\lim_{x\to 0}g(x)=\frac{sin x}{x}=1$$  
This means that the function value approaches $1$ as $x$ tends to $0$. This doesn't mean that the value of the function at $x=0$ is $1$. Remember we cannot remove a permanent singularity by using limits. What we will get is the neighbouring value around $x=0$. In some functions, unlike what we have seen earlier, there is no of convergence. In such cases, we say the function has no limit at that particular point.  

How this limit can be used to explain instantaneous velocity?  

Well, we cannot directly measure instantaneous speeds. You need a small length $\Delta x=x_2-x_1$ and a stopwatch. You start the stopwatch when the car enters the point $x_1$ and stop when it leaves the region $x_2$. Of course, this is not the velocity at that particular instant. We can measure an event in a time interval, here the time interval is $\Delta t=t_2-t_1$. However we choose $\Delta x$ to be as small (but measurable too) as possible. Now, we do the math. To get the instantaneous speed around a particular time within the the interval $\Delta t$, say $t$, such that $t_1<t<t_2$, we are going to shrink the time interval $\Delta t$ around the time $t$. The instant $t$ hence corresponds to the situation, when the width of the interval around $t$ reaches zero. However, in that case, we have only one position at a particular time &t&. To measure velocity, we need a change in position. But, there exists a universal speed limit. So, we search out to find the displacement for $\Delta t\to 0$, assuming that the width of the time interval lies around $t$. Hence instantaneous speed is defined as  
$$ \bbox[5px,border:2px solid red]
{
v(t)=\lim_{\Delta t\to 0}\frac{\Delta x}{\Delta t}
}
$$  
The equation means that to find the instantaneous speed of a particle (i.e., the speed of the particle at any instant $t$: $v(t)$), we take a small displacement the particle travelled in a time interval $t$ and then find out the velocity with the time interval approaching zero around $t$.  
This equation is generally valid for any real life situations. But when you divide displacement by time, you will get only the average velocity, which is not in general equal to the velocity of the particle at some instant within the time interval. Average is like assuming everything as equal and distribute what you have equally to everyone. Differentiating (which is an extend to limits) is like giving everyone in accordance with their need. Which one you see as better way of distribution?  
A: $\delta s$ is just a way to write a distance, $\delta s=s_{end}-s_{start}$, and same for time, $\delta t=t_{end}-t_{start}$. Often $\Delta$ is used instead of $\delta$, but that doesn't matter; they are just symbols invented to mean difference.
When you do distance over time, $v=\delta s/\delta t$, it will always be an average:


*

*Going several kilometers to work, the whole distance over the duration gives an average speed.


But you might have driven uphill and downhill and stopped at red lights along the way:


*

*So instead, say distance over time for just the uphill-part and then the downhill-part etc. Different parts of the trip have different speeds then. But still average speeds.


Continue to split the trip into more and more parts:


*

*Calculate the speed for each meter, and you get a much better picture of the speeds along the way. Still only average speeds, but many average speeds over shorter distances (and times).


And now the idealization:


*

*Imagine splitting into infinitely many parts. So small that they are almost negligibly short and only lasting a blink of an eye - only a moment in time. The speeds would still be averages, but averages over basicly just a point (a point in time and a point-like distance).


In that case we can choose to call them instantaneous, because that is basicly what they are; instantaneous speeds taking place at just a point.  And we can choose to invent a specific symbol for them, for example by changing the $\delta$ to a $d$:
$$v=\frac{ds}{dt}$$
So, when $\delta t$ becomes smaller and smaller and smaller and almost reaches 0, then we call it instantaneous and replace $\delta$ with $d$. This is the limit. Of course $\delta t$  can't become exactly zero (then we wouldn't have moved at all), but it can come damn close. That is the limit. And someone (G. Leibniz, I believe) invented a mathematical way of writing such a limit:
$$v=\lim_{\delta t→0} \frac{\delta s}{\delta t}=\frac{ds}{dt}$$
which in words just reads: "Speed is distance over time when that time goes towards it's limit at 0. And then we call it $ds/dt$ instead".
