What is a limit? Why study them?

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Learning math requires reading a lot. One idea seems to endlessly lead to another and it is understandable to ask: where is all this going? And so, like any such mathematical exposition, I must ask you to bear with me.

Although the connections between concepts are not always immediately clear, they always seem so when you review the ideas later. As a result, a few layers of context must be established to answer the questions: what is a limit and why are limits studied?

A limit must be understood in the context of a function.

A relation is an operation that accepts an input and yields an output. A function is a certain kind of relation where every member of the input maps to only one member of the output. For example, in the function [katex]f(x) = x + 3[/katex], replacing [katex]x[/katex] with 2 will always result in 5, and never any other number.

The members of the input of a function are in a set that is called the domain while the members of the output of a function are in the range.

A limit is a value that is approached in the function’s range as a number in the domain approaches a certain value, but the limit does not need to be an actual member of the range.

This concept is very much akin to how we generally use the word “limit”. On the one hand, there is a speed limit when driving, where we are expected to only go a certain limit and not too much above it. But the mathematic concept is more connected to very hard limits that precisely denote what value is being approached in the range as the corresponding value is approached in the domain. Think of 100°C as the limit at which water boils. Also consider cutting a sheet of paper in half and continually cutting the pieces in half until you reach indivisible units, but in an ideal circumstance you can continue cutting

Most of calculus springboards off a very particular application of a limit, but a significant amount of the topic must be dedicated to proving that limits are concrete things on which arithmetic operations can be performed, even if the limit corresponds to a value that is not in the range of the function. This particular attribute is very important in regard to the primary way that limits are applied in understanding a function’s rate of change.

A function’s rate of change is the slope of every point of a function. With a linear function, the slope of every point in the function is clearly defined. But how would you calculate the rate of change of a single point on a curve? This is where limits shed significant light on the situation.

Imagine driving a car. In this car you are traveling for a certain distance and for a certain period of time. We can take sections of the distance you traveled along with the corresponding time to find out the speed traveled in that distance and time. We can create a distance function where the input is a point in time which maps to the specific distance traveled at that point in time.

The distance formula can be written as [katex]s(t)[/katex]. To find the average velocity of your trip in a certain period of time, we divide the difference of your distance (displacement) by the amount of time that passed:

v_{av} = \frac{s(a)-s(t)}{a-t}

Where [katex]a[/katex] is the time at which we stopped measuring average velocity and [katex]t[/katex] is the time at which we began. This can also be written as:

v_{av} = \frac{s(t+h)-s(t)}{t+h-t}

Which shows how the time at which we stopped measuring for average velocity, [katex]t+h[/katex], is at “start time plus amount of time that passed” and [katex]t[/katex] is the time at which we started measuring. Note how the denominator can be simplified to:

v_{av} = \frac{s(t+h)-s(t)}{h}

If our distance formula was a straight line, then this ratio would be the same everywhere, because it is precisely the measurement of slope on a graph. In that case, the average velocity between two distant points would be the exact same as the instantaneous velocity at any single point on a straight line. For a curved graph, this formula would only give you an average, with all the points in a curve between [katex]t[/katex] and [katex]t+h[/katex] having varying slope.

But if you took this average velocity formula and were to make [katex]h[/katex] small enough so that it was practically zero so as to make as accurate of an average as you could so that you would ideally see a straight line if you zoomed into the curve close enough, then you would start to approach a value for this function that is in fact the instantaneous rate of change (slope) at point [katex]t[/katex]:

v_{in} = \lim_{h\to0}\left(\frac{s(t+h)-s(t)}{h}\right)

Streamlining the processes for finding the slope of tangent lines on any point of a function has endless applications. Calculating the rate at which some functions change in relation to others allows you to make powerful predictions that previously seemed difficult or impossible to calculate. There are numerous applications of limits in marketing and economics, such as estimating costs of production and determining the elasticity of the market. In the graphic arts/drafting world, the ability to fine tune the rate of change of curves in a piecemeal fashion allows you to draw any shape conceivable. In general, the study of limits provides a deeper understanding of how functions relate to each other than previously understood, so that several correct assumptions about, or many streamlined solutions for, complex problems can be made.

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