What Is a Limit in Calculus and How Does It Work?

TL;DR

A limit assigns formal notation to the intuitive idea of one value approaching another, letting calculus avoid talking about infinitely small changes. The formal derivative is written as the limit, as h approaches 0, of f(x+h) minus f(x) all over h, where h is an ordinary finitely small number like 0.001. The epsilon-delta definition makes 'approach' rigorous.

Transcript

The last several videos have been about the idea of a derivative, and before moving on to integrals I want to take some time to talk about limits. To be honest, the idea of a limit is not really anything new. If you know what the word approach means you pretty much already know what a limit is. You could say it's a matter of assigning fancy notatio... Read More

Key Insights

• A limit is nothing conceptually new: if you know what the word approach means, you already know what a limit is. It is a matter of assigning fancy notation to the intuitive idea of one value getting closer to another.
• The derivative is not the ratio df/dx itself but whatever that ratio approaches as dx approaches 0. The ratio is the rise-over-run slope between the starting point on the graph and the nudged point, which is almost, but not exactly, the derivative.
• Inside a limit expression you almost never see lowercase-d terms like dx. The standard is a different variable such as delta x, or commonly h, because terms with a lowercase d already have the limiting process built into them.
• The value h in the formal definition is the exact same thing as dx: a nudge to the input of f with some non-zero, finitely small size like 0.001. The definition analyzes what happens for arbitrarily small choices of h.
• The expression (2+h) cubed minus 2 cubed, all divided by h, is what pops out of the derivative definition for x cubed at x equals 2. It is undefined at h equals 0, giving 0 divided by 0, so its graph has a hole there, yet the limit is 12.
• A limit exists when shrinking the range of input values around the limiting point forces the corresponding range of output values to close in on one value, and that output range can be made as small as you want.
• A limit fails to exist when the function approaches different values from each side. In the counterexample that jumps at 0, approaching from the right gives 2 and from the left gives 1, so the output range never shrinks smaller than 1.
• The function sin(pi times x) divided by (x squared minus 1) is undefined at x equals 1, since sin of pi is 0 and the denominator is 0. Plugging in 1.00001 approximates the limit as a number around negative 1.57.

Questions & Answers

Q: What is the formal definition of a derivative?

The formal definition writes the derivative as the limit, as h approaches 0, of f at the starting input plus h, minus f at the starting input, all divided by h. You start by imagining nudging the input some little amount away and looking at the resulting change to the output. The ratio of that output change to the input change is the rise-over-run slope between the starting point on the graph and the nudged point, and the actual derivative is whatever that ratio approaches as the nudge approaches 0.

Q: Why do textbooks use h instead of dx in the definition of a derivative?

The value h is the exact same thing as dx: a nudge to the input of f with some non-zero, finitely small size, such as 0.001. The only reason people introduce a new variable name into the formal definition rather than just using dx is to be extra clear that these changes to the input are ordinary numbers that have nothing to do with infinitesimals. You also almost never see lowercase-d terms inside a limit expression, since terms like dx already have the idea of a limit built into them. Other common choices are delta x, or commonly h.

Q: What is the epsilon-delta definition of a limit?

Think about any distance away from the limiting output value, denoted by the Greek letter epsilon, where the intent is that epsilon is as small as you want. For the limit to exist, you must always be able to find a range of inputs around the limiting point, some distance delta around it, so that any input within delta corresponds to an output within a distance epsilon of the limiting value. The key point is that this holds for any epsilon, no matter how small, you will always be able to find the corresponding delta.

Q: How do you know when a limit does not exist?

A limit does not exist when there is no single clear, unambiguous value the function approaches. In the video's counterexample, a function undefined at 0 that jumps at that point approaches 2 from the right and 1 from the left. When you shrink the input range around 0, the corresponding outputs do not narrow in on any specific value; they straddle a range that never shrinks smaller than 1. In epsilon-delta terms, you can find a sufficiently small epsilon, like 0.4, so that no matter how tiny delta is, the corresponding output range is always too big.

Q: Why do limits let calculus avoid infinitely small changes?

The big fuss about limits is that they let us avoid talking about infinitely small changes by instead asking what happens as the size of some small change to our variable approaches 0. Nothing about the right-hand side of the formal derivative definition references the paradoxical idea of an infinitely small change; the point of limits is to avoid that. The nudge h always has a non-zero, finitely small size, and the definition simply analyzes what happens for arbitrarily small choices of that size.

Q: Why does the graph of (2+h)^3 minus 2^3 divided by h have a hole at h equals 0?

Plugging in h equals 0 gives 0 divided by 0, which is not defined, so the function has no value at that single point. The graph is drawn with an exaggerated empty circle to mark the hole. Everywhere else the graph looks like a nice continuous parabola, which makes sense because it is a cubic term divided by a linear term. Importantly, the function is perfectly well defined for inputs as close to 0 as you want, and as h approaches 0 from either side, the height of the graph approaches 12.

Q: How should you interpret dx and df when learning calculus?

You can and should interpret dx as a concrete, finitely small nudge, just so long as you remember to ask what happens when that thing approaches 0. Some people instead interpret dx as an infinitely small change, or say dx and df are nothing more than symbols not to be taken too seriously, but the video favors neither view. Treating them as concrete finite nudges builds stronger intuition for where the rules of calculus come from, and it is not merely an intuition trick: everything said with that philosophy is just a translation of the formal definition.

Q: How can you estimate the limit of sin(pi x) divided by x squared minus 1 at x equals 1?

At x equals 1 the function is not defined, because sin of pi is 0 and the denominator x squared minus 1 also comes out to 0, so the graph has a hole there. The same thing happens at x equals negative 1. One way to approximate the limit is to plug in a number that is just really close to 1, such as 1.00001, which gives a number around negative 1.57. To find the value precisely rather than approximately, derivatives can come back and return the favor, which is the idea behind L'Hopital's rule.

Summary & Key Takeaways

• The formal definition of the derivative writes df/dx as the limit, as h approaches 0, of the quantity f at the starting input plus h minus f at the starting input, all divided by h. Terms with a lowercase d already have the idea of a limit built into them, so df/dx is shorthand for that fuller expression.

• Nothing in the formal definition references an infinitely small change, and the whole point of limits is to avoid that paradoxical idea. The variable h is the exact same thing as dx, a non-zero finitely small nudge; a new variable name is introduced only to make clear these are ordinary numbers, not infinitesimals.

• The epsilon-delta definition says a limit exists when, for any distance epsilon around the limiting output no matter how small, you can find some distance delta around the limiting input so every input within delta maps to an output within epsilon. When no limit exists, some epsilon like 0.4 defeats every delta.