calc4phys: limits and continuity (+ handout)
a brief intro to calc so you can do calc-based physics :)
I’ve written previously about how learning calc completely transformed my understanding of physics. Calc may seem like an intimidating subject, but the calc needed for physics is learnable in a few days.
Find the neatly formatted PDF handout here (with definitions, theorems, tricks, good notation, and sample problems):
I recommend that you take a look at the above handout instead of just reading this hodgepodge of a blog post. Unfortunately, Substack does not support inline Latex at this time, so I will only be going over a conceptual overview in the rest of this post.
Calc4Phys Lesson One: Limits and Continuity
Limits form the basis for just about everything in introductory calculus.
Given a function f, the limit of f(x) as x approaches c is a real number L if f(x) can be made arbitrarily close to L by taking x sufficiently close to c.
In essence, taking the limit of f(x) as x approaches some number c means we’re moving x super, super close to c. In most graphs that you’re familiar with, we find that x ends up equaling c. But not always!
In the graph above, as x approaches -1 from the left side (we call this a left-sided limit), we seem to reach y=1. But as x approaches -1 from the right side (we call this a right-sided limit), we seem to instead reach y=2. Hmm. Weird. When the left-sided and right-sided limits don’t equal each other, we say the limit does not exist.
Since the bottom dot is shaded, we know f(-1) is defined at y=1. This is a case where the limit at x=c (as we move x super close to -1 from both sides) does not exist, but the function is still defined at x=c. Cool!
Usually, we can evaluate limits by just plugging in a value. The limit of f(x)=x as x approaches 3 is just f(3)=3. Easy! This method is called direct substitution, and it should always be your go-to.
Sometimes, though, direct substitution gives us a funky result like 0/0. This is called indeterminate form. For now, this means we have to manipulate the function into a different form and then try directly substituting again. (We’ll bypass this later using a satisfying trick called L’Hospital’s Rule.)
Another funky result is z/0 where z is nonzero. This means we have a vertical asymptote, and we need to evaluate the right-sided limit and the left-sided limit.
The especially interesting thing about limits goes far beyond these concepts. For instance, you’ve probably heard of these things called “derivatives” which are related to slope. Lots of physics resources refer to “instantaneous accelerations” and “derivatives” together. It seems like a derivative is some kind of “instantaneous rate of change.”
What does it mean to be instantaneous?
This is a question that may have hypothetically haunted me for years and scared me away from calculus. But I’m confident you can take a shot at answering it.
Think of how you would calculate the rate of change for two points on any graph. Rise over run, delta y over delta x—whichever slope slogan floats your boat. Now, imagine the points getting very, very close. The x-coordinates of each point are approaching each other. They’re infinitesimally close. Hmm. This language sounds familiar.
Let’s give this idea numerical representations. Say one point has coordinates (x, f(x)), and the other has coordinates (x+h, f(x+h)). The slope between these two points is (f(x+h)-f(x))/h. When these points get super close to each other, the difference h between their x-coordinates approaches 0. This reeks of limits!
It’s your turn to take these ideas and run with them. Try to derive the slope between two infinitesimally close points (x, f(x)) and (x+h, f(x+h)) using limits. This is called the instantaneous rate of change, and it’s also known as the limit definition of the derivative. Feel free to search up this definition (no spoilers here!) and see if your answer matches up.
The next handout will officially begin to segue into derivatives with a formal explanation and proper notation.
Play around with my intro to derivatives Desmos graph if you want to peek ahead: https://www.desmos.com/calculator/0f5ekprz6w.
If you want more detail on limits, here’s a reminder to check out the PDF: