Teaching Notes · Math for Social Sciences

Lines, Limits, and Continuity

A straight line is the simplest relationship between two variables, and it turns out to already contain the idea that the rest of calculus builds on: a rate of change that stays constant. This set of notes starts with lines, where that rate is just the slope, then moves to limits, where the idea of a rate of change that is allowed to vary finally gets a precise definition, and closes with continuity, the condition a function needs to satisfy before any of the later machinery can be safely applied to it.

Slope as a business rate

The equation y = mx + c describes every straight line, with c giving the point where the line crosses the y axis and m giving the slope, the number of units the line rises or falls for each unit moved horizontally. A manufacturing example makes the two constants concrete. If the total cost of producing x units is C = 25x + 3500, the y intercept (0, 3500) is the cost of producing nothing at all, the fixed cost that has to be paid regardless of output. The slope, 25, is the cost added by each additional unit, what economists call the marginal cost. Read this way, a line is not just a graph; it is a cost structure, split cleanly into a fixed part and a per unit part.

Given any two points on a line, (x1, y1) and (x2, y2), the slope between them is m = (y1 − y2)/(x1 − x2). A line through (−1, 2) and (2, 2) has slope (2−2)/(2−(−1)) = 0, a horizontal line. A line through (2, 5) and (2, 1) has an undefined slope, since the two points share the same x coordinate: a vertical line has no slope at all. Once the slope and one point are known, the full equation follows from the point slope form, y − y1 = m(x − x1). A line of slope 3 through (1, −2), for instance, works out to y + 2 = 3(x − 1), or y = 3x − 5.

Put both ideas together and a two point word problem becomes routine. A company builds a warehouse for 825,000 that is expected to be worth only 75,000 after 25 years of use, and assuming the value declines along a straight line, the value y at time t (in years) passes through (0, 825000) and (25, 75000). The slope is (75000 − 825000)/(25 − 0) = −30000, so the value over time is given by y = −30000t + 825000, a single equation that reads off the asset's value at any point across its useful life, straight line depreciation in its simplest form.

What a limit actually means

A limit asks a narrower question than the value of a function at a point: it asks what the function is doing as the input gets arbitrarily close to that point, whether or not the function is even defined there. Take f(x) = x² − x + 2 and ask what happens as x approaches 2. Checking values on both sides, 1.9, 1.95, 1.99 on one side and 2.1, 2.05, 2.01 on the other, the outputs close in tighter and tighter on 4 from both directions. That shared value is the limit, written lim(x→2) f(x) = 4, and for a polynomial it is always just the value of the function at that point, found by direct substitution.

The more interesting case is a function that is not even defined at the point in question. Consider f(x) = (x² − 1)/(x − 1). Substituting x = 1 gives 0/0, undefined, so f has no value at x = 1 at all; its graph has a hole there. But for any x not equal to 1, the expression simplifies, since x² − 1 factors as (x − 1)(x + 1), leaving f(x) = x + 1 for every other input. As x approaches 1 from either side, f(x) approaches 2, so the limit exists and equals 2 even though the function itself is undefined exactly at that point. This is the key distinction a limit captures: a function's behavior near a point does not have to match, or even exist at, the function's value at that point.

Handling the awkward cases: the replacement theorem and the root trick

Direct substitution works whenever a function is a polynomial, or a rational function whose denominator does not vanish at the point in question. When substitution produces 0/0, as it did above, the standard fix is to factor both the numerator and the denominator and cancel the shared factor, a technique justified by what the notes call the replacement theorem: if two functions agree everywhere except possibly at one point, they have the same limit at that point. For lim(x→−3) (x² + x − 6)/(x + 3), direct substitution gives 0/0, but factoring the numerator as (x + 3)(x − 2) and cancelling the shared (x + 3) leaves lim(x→−3) (x − 2) = −5.

A related technique, sometimes called the root trick, handles limits where a square root produces the same 0/0 problem. For lim(x→0) (√(x + 4) − 2)/x, multiplying numerator and denominator by the conjugate √(x + 4) + 2 clears the root from the numerator:

[(x+4) − 4] / [x(√(x+4)+2)] = x / [x(√(x+4)+2)] = 1 / (√(x+4)+2)

which, substituting x = 0, gives 1/4. The trick is always the same move: multiply by a conjugate to turn a difference of roots into a difference that cancels the troublesome x in the denominator.

One sided limits

Some functions, especially ones defined by different formulas on different intervals, approach different values depending on which side you approach from. The limit from the left, written lim(x→a⁻) f(x), and the limit from the right, lim(x→a⁺) f(x), need to agree before the ordinary two sided limit can be said to exist at all. Consider a function defined as f(x) = 2x + 1 for x below 0, and f(x) = 4x² − 2 for x above 0. Approaching 0 from the left uses the first formula and gives 2(0) + 1 = 1; approaching from the right uses the second and gives 4(0)² − 2 = −2. Since 1 does not equal −2, the limit of this function as x approaches 0 simply does not exist, even though both one sided limits exist individually.

Continuity: when a graph has no breaks

A function f is continuous at a point a when three conditions all hold at once: f(a) is actually defined, the limit of f(x) as x approaches a exists, and that limit equals f(a) itself. If any one of the three fails, the function is discontinuous there, and the type of failure usually has a visible shape on the graph. The function (x² − 1)/(x − 1) from earlier is undefined at x = 1, so it fails the first condition outright, which shows up as a hole in its graph exactly at that point.

A different kind of break shows up when the one sided limits disagree. Take a function defined as f(x) = x/2 − 1 for x at most 2, and f(x) = x + 1 for x above 2. At x = 2 itself, f(2) = 2/2 − 1 = 0, so the function value exists. But the limit from the right, using x + 1, gives 2 + 1 = 3, while the limit from the left, using x/2 − 1, gives 1 − 1 = 0. Since 3 does not equal 0, the two sided limit at x = 2 does not exist, so the function is discontinuous there, and its graph visibly jumps rather than having a simple hole.

A third kind of discontinuity can occur even when every piece looks fine on its own. Suppose f(x) = 2 exactly at x = 1, and f(x) = 4x + 1 for every other x. Here f(1) = 2 is defined, and the limit as x approaches 1 is lim(x→1) (4x + 1) = 5, which does exist. But 5 does not equal f(1) = 2, so the third condition fails, and the function is discontinuous at x = 1 even though nothing about its graph looks broken from a distance, aside from a single relocated point.

Two broader rules save a lot of point by point checking. Every polynomial is continuous at every real number, with no exceptions to check for. Every rational function is continuous everywhere on its domain, meaning everywhere except at the zeros of its denominator; the function (x² + 2x)/(x² − 25), for example, is continuous on every real number except 5 and −5, the two points where its denominator vanishes.

Continuity is not just a technical housekeeping condition. Every tool that follows in this course, starting with the derivative itself, is built on functions that behave predictably near a point, and continuity is exactly the condition that rules out the holes and jumps that would make those tools meaningless.

Source: lecture notes for an introductory calculus course for the social sciences, covering lines and slope, the limit of a function and the rules for evaluating limits, the replacement theorem, the root trick, one sided limits, and continuity.