How Does Limit Notation Solve The Problem Of Instantaneous Velocity?

Question

I am struggling to see how limit notation solves the problem of instantaneous velocity. Because limit notation at best gives the velocity over a timeframe that is infinitely close to 0, basically the velocity at an infinitely small time period. But it cannot give the velocity at a specific point, because that would lead to a denominator of 0 and an answer of undefined. So how does limit notation solve the problem of instantaneous velocity?

Answer 1

A conveyor belt has a constant velocity which you can easily measure using a tape measure and a stopwatch.

On the other hand, on the intuitive level, the velocity of an accelerating car is surely increasing with time and in no small time interval constant. If we shoot a picture of this car we have a precise record of its location within the surroundings at the moment of clicking, but absolutely no idea of its speed. Maybe the car was not even running.

Now we could draw (or have automatically drawn) a length $s(t)$ vs. time $t$ graph of the movement of this car. We would then see a parabola like convex curve. For a constant speed car the curve would be a straight line, and the slope of this line encodes the speed of this car. From an intuitive standpoint it is then natural to call the slope of the tangent to the parabola at a given point $\bigl(t,s(t)\bigr)$ the momentary speed of the car at that point of time. This would be an acceptable precalculus definition.

But you are a student now; and you have learned that there is an exact mathematical concept covering this idea, namely the limit $$v(t):=\lim_{\Delta t\to0}{s(t+\Delta t)-s(t)\over\Delta t}\ .$$ This is a definition and replaces prior handwaving about "infinitesimally small time periods".

Answer 2

Your objection applies more generally to the definition of a tangent line as a limit. Do you doubt that a smooth curve like a circle or a parabola should have a tangent line at each point? While you focus on the denominator at the point being $0$, you ignore that the numerator is also $0$ and that is crucial.

For example, consider $f(x) = (\sqrt{x+1}-1)/x$. At $x=0$ the numerator and denominator are both $0$ and the ratio at $x=0$ does not make sense. However, there is a sensible trend in the values if you look at $f(x)$ for $x$ near but not equal to $0$, e.g., $f(.001) =\approx .49987$ and $f(-.001) \approx .50012$. We can rewrite $f(x)$ for $x \not= 0$ to explain this: $$ \frac{\sqrt{x+1}-1}{x} = \frac{1}{\sqrt{x+1}+1} $$ and for $x$ close to $0$ the right side gets close to $1/(1+1) = 1/2$.

We formalize the trend in values around a point in place of a value at the point (which might not make sense) by speaking of the limit at that point. Limits take us beyond the idea of direct substitution as in algebra, and dealing the transition from computation by substitution (algebra) to computation by limits (calculus) is a common source of difficulty by students, but the entire machinery of calculus and calculus-based applications (in physics, engineering, etc.) is based on this limit idea. The creators of differential calculus (Newton and Leibniz, and Fermat before them) did not speak in terms of limits and they encountered criticism somewhat like the type you are bringing up (thinking in terms of substitution, like algebra). It took a long time for the foundations of calculus to be made solid.

Any time you meet a ratio with a meaningful limit value and the denominator is tending to $0$, the numerator is tending to $0$ also. Just because a nonzero constant divided by something tending to $0$ will become very large (in absolute value) does not mean a ratio of two expressions that both tend to $0$ can't have a sensible finite limit.

By the way, the premise of your question is mistaken: notation is not solving a problem, but rather an idea is solving a problem, namely the idea of a limit is allowing us to make sense of what, purely algebraically, seems like something meaningless. This happens throughout mathematics. If you refuse to expand your mind to accept that new ideas can let us speak about things that at first appear "meaningless" in your original more narrow mindset then you won't get far. For example, you will never understand how we can explore and work in spaces of dimension greater than $3$, or even in infinite-dimensional spaces, if you stubbornly assert that such concepts are meaningless because they don't make direct intuitive sense due to the lack of pictures in higher dimensions.

Answer 3

It can be misleading to think of the limit concept in an operational way. You don't really "take a limit," as one might hear people say. The notation $\lim_{h\to 0}g(h)=c$ represents a promise that no matter how close you want $g(h)$ to be to $c$, there is always some (possibly very small) interval of values for $h$ around $0$ to achieve this closeness (possibly excluding $h=0$).

If you are able to write $\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=r$ for some $x$ and some $r$, that is a promise that no matter how close you want the slope of a secant line from $(x,f(x))$ to be to $r$, there's some (possibly very small) interval of values for $h$ around $0$ to achieve such slopes.

That is, the derivative $f'(x)=r$ is a slope that can be well-approximated to an arbitrarily good degree by slopes of secant lines. Our intuition is that this is the slope that secant lines "converge to."

We have decided as mathematicians and physicists that this is the definition of instantaneous velocity. A secant line slope corresponds to the velocity estimate you get by measuring time and distance at two points in time. An instantaneous velocity is a promise that by measuring with ever smaller $\Delta t$'s, your estimates would get closer and closer to this supposed "instantaneous" velocity. Since this is a math site, I'll have to leave it to the philosophers about the reality of instantaneous velocity beyond this.

Limits also solve the problem of position. I have heard it called Newton's zeroth law that the positions of particles follow continuous trajectories. A function is continuous if $\lim_{x\to c}f(x)=f(c)$ for all $c$. This means that $f(x)$ can be made to be as close as you want to $f(c)$ so long as $x$ is anything close enough to $c$. So, for Newtonian physics, the position of a particle at a particular time can be pinned down to arbitrary precision with sensitive enough time measurements.

My understanding is that continuity resolves Zeno's racetrack paradox (that a runner can't possibly reach the end of the racetrack, because there will always be half left), and instantaneous velocity resolves Zeno's arrow paradox (that an arrow cannot possibly be moving because if you look at it in an instant, you cannot distinguish it from an arrow that is not moving). For the racetrack, since the times at which the runner reaches the half-remaining marks have a limit, by continuity of position one gets that the runner's position at that limit is the limit of positions at those times, which is the end of the racetrack. For the arrow, instantaneous velocity is defined in a way that depends on the behavior of the arrow for an entire (small) time interval. (That is, it's resolved by saying "whatever -- we can just sidestep the issue about whether velocity is a property of a particular moment by instead defining it in a way that depends on a continuum of moments. And, if it turns out instantaneous velocity is physically real, our definition must provably be equivalent.")

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Warning: borderline philosophy.

Sometimes people say that velocity at a particular point in time is meaningless. It is as meaningless as talking about the position of a particle at a particular point in time. Or even a particular point in time. This comes down to the issue of what a real number even is, and what mathematicians have settled on is, roughly, that a real number is something that can be approximated by rational numbers. Rational numbers represent what we can actually measure. A point in time is sort of the idea that if we could get ever more precise clocks, we would be able to specify points in time with greater and greater precision. A similar thing goes for points in space and ever more precise rulers. There is some physical evidence that neither of these assumptions are valid (Planck time and Planck length), but worst case real numbers are a useful fiction. Instantaneous velocity is sort of the idea that if we used extremely precise clocks and extremely precise rulers, on small enough timescales trajectories look more and more indistinguishable from constant-velocity trajectories.

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In short, limits give meaning to the phrase "arbitrarily/infinitely close to $0$," which, if you don't have a precise definition, what would you even mean by that?

In math, things gain meaning based on what you can do with them. We use precise definitions because you can do more with them. That is, they are more meaningful. If the concept of a limit could be so easily slain by saying "but since $h$ is getting infinitely close to $0$, the denominator is $0$ so the secant line would have no defined slope," then we would have a bad definition of a limit, and we'd need to find a better one, if we could (and we have).