Difficulty: Introductory | Prerequisites: Algebra II, basic function notation, slope of a line.
These two sections launch the core idea behind all of calculus: measuring how fast something changes at a single instant. Section 4 builds from average rates of change (secant lines) toward instantaneous rates (tangent lines). Section 5 formalises that "getting closer" process with the concept of a limit, including one-sided limits, infinite limits, and vertical asymptotes. If you are joining the course late, start here; everything in Chapter 2 rests on this foundation.
Calculus begins with one question: how fast is something changing at a single moment? Secant lines measure the average rate of change between two points; tangent lines measure the instantaneous rate of change at one point. To make "instantaneous" precise, we need limits, which describe what value a function approaches as the input gets arbitrarily close to some number.
Secant line
A line that passes through two points on the graph of a function. Its slope equals the average rate of change of the function over that interval.
In simple terms, draw a straight line connecting two points on a curve. The steepness of that line tells you, on average, how fast the function was changing between those two points.
Tangent line
A line that just barely touches ("skims") the graph of a function at a single point, matching the curve's direction there. Its slope equals the instantaneous rate of change at that point.
Think of it as the secant line when the two points are squeezed infinitely close together. It captures how fast the function is changing at one specific moment.
Average rate of change (ARoC)
The ratio [f(x₂) - f(x₁)] / [x₂ - x₁], representing the slope of the secant line between the points (x₁, f(x₁)) and (x₂, f(x₂)). Units are always units of f divided by units of x.
In simple terms, this is the "overall speed" between two moments. If you drove 90 miles in 2 hours, your average rate of change of distance was 45 mph.
Instantaneous rate of change
The rate at which a function is changing at one precise input value. Geometrically, it is the slope of the tangent line at that point.
Think of it as the number on your speedometer at a single instant, not your average speed over a whole trip.
Limit (lim)
We write lim(x→a) f(x) = L to mean: the values of f(x) can be forced as close to L as we like by requiring x to be sufficiently close to a (but not equal to a).
In simple terms, "limit" answers the question: what value does f(x) approach as x gets closer and closer to a? The function need not actually equal L at x = a.
Left-hand limit
Written lim(x→a⁻) f(x) = L. The value f(x) approaches as x comes toward a from values less than a (from the left on the number line).
Right-hand limit
Written lim(x→a⁺) f(x) = L. The value f(x) approaches as x comes toward a from values greater than a (from the right on the number line).
Vertical asymptote
The line x = a is a vertical asymptote of y = f(x) if at least one of the one-sided limits at a is +∞ or -∞. Graphically, the curve shoots upward or downward without bound near x = a.
In simple terms, the function "blows up" near that x-value. You cannot plug that value in, and the graph goes off toward infinity.
Infinite limit
A limit where f(x) grows without bound (positively or negatively) as x approaches some value a. Written as lim(x→a) f(x) = ∞ or lim(x→a) f(x) = -∞. Technically the limit does not exist as a real number, but the notation tells you the specific way it fails.
A secant line connects two points on a curve: P(x₁, f(x₁)) and Q(x₂, f(x₂)).
Slope of the secant line = [f(x₂) - f(x₁)] / [x₂ - x₁]. This is the average rate of change on the interval [x₁, x₂].
The units of ARoC are always (units of output) / (units of input). If f gives distance in miles and x is time in hours, ARoC is in miles per hour.
Example pattern: for f(x) = √(x + 1), the ARoC on [3, 8] is [f(8) - f(3)] / [8 - 3] = [3 - 2] / 5 = 1/5.
A tangent line touches the graph at exactly one point and "skims" the curve there.
You approximate the tangent line's slope by computing secant-line slopes with the two points very close together.
As the second point slides toward the first, the secant slopes approach the tangent slope. This limiting process is the bridge to calculus.
Formally, the slope of the tangent line at x = a is found by computing the slope of the secant through (a, f(a)) and (a + h, f(a + h)), then letting h approach 0.
The slope of the tangent line = the instantaneous rate of change at that point.
Example pattern: for f(x) = x² - 3 at x = 2, build the secant through (2, f(2)) and (2 + h, f(2 + h)), simplify, let h → 0. You find the slope is 4.
For f(x) = x² + 4x, to estimate the slope at x = 4:
Compute the secant slope through (4, f(4)) and (4 + h, f(4 + h)).
Simplify the difference quotient: [f(4 + h) - f(4)] / h.
As h gets small, the expression approaches the tangent slope (in this case, 12).
Then the tangent line equation uses point-slope form: y - f(4) = 12(x - 4).
lim(x→a) f(x) = L means: f(x) can be made as close to L as desired by taking x sufficiently close to a (but x ≠ a).
The value of f(a) itself does not matter for the limit. The function might not even be defined at a.
Left-hand limit: lim(x→a⁻) f(x) = L, approaching from values less than a.
Right-hand limit: lim(x→a⁺) f(x) = L, approaching from values greater than a.
The two-sided limit lim(x→a) f(x) = L exists if and only if both one-sided limits exist and are equal: lim(x→a⁻) f(x) = L and lim(x→a⁺) f(x) = L.
For piecewise functions, check each piece separately on its side of the boundary.
When reading limits from a graph, look at where the curve heads, not where the dot sits. An open circle at a point means the function is not defined there (or defined differently), but the limit can still exist.
A vertical asymptote occurs at x = a when the function's values grow without bound near a.
For functions of the form g(x)/h(x), if g(a) ≠ 0 and h(a) = 0, there is often a vertical asymptote at x = a.
To confirm and determine the sign of the infinity, check the left-hand and right-hand limits separately by plugging in values slightly less than and slightly greater than a.
Example: for (x² + 2x + 1) / (x + x²), factor the denominator as x(x + 1). Vertical asymptotes occur at x = 0 and x = -1 (where the denominator is zero but the numerator is not).
When evaluating an infinite limit, think about whether the numerator is positive or negative, and whether the denominator is a small positive or small negative number. A positive divided by a small positive gives +∞; a positive divided by a small negative gives -∞.
\text{Slope of secant line} = m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\text{Slope of tangent line at } x = a \;\colon\quad m = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\lim_{x \to a} f(x) = L \quad \Longleftrightarrow \quad \lim_{x \to a^-} f(x) = L \;\text{ and }\; \lim_{x \to a^+} f(x) = LVertical asymptote detection: for a rational function g(x)/h(x), check where h(a) = 0. If g(a) ≠ 0 at that same point, x = a is a vertical asymptote. If both g(a) = 0 and h(a) = 0, factor and simplify first (this is the 0/0 case covered in Section 6).
Average and instantaneous rates of change appear everywhere outside the maths classroom. A speedometer reads instantaneous velocity (tangent-line slope), while "we averaged 60 mph on the trip" is a secant-line slope. Economists use average rate of change to describe how sales or populations grew over a period, and the instantaneous rate to pinpoint when growth peaked. Infection-rate modelling, chemical reaction rates, and acceleration in physics all rest on exactly these ideas.
Students often confuse the value of f(a) with lim(x→a) f(x). The limit depends on what happens near a, not at a. A function can have a limit of 5 at x = 2 even if f(2) = 7 or f(2) is undefined.
Students sometimes assume that if a limit equals ∞ then the limit "exists." Technically, an infinite limit means the limit does not exist as a real number. Saying lim = ∞ is a specific way of describing that failure.
Forgetting to check both one-sided limits before claiming a two-sided limit exists. If the left and right sides disagree, the two-sided limit does not exist, even if each side individually approaches a finite number.
Confusing a vertical asymptote (function blows up) with a hole (removable discontinuity). A hole occurs in the 0/0 case after cancellation; a vertical asymptote occurs in the N/0 case.
⚠️ The difference quotient [f(a+h) - f(a)] / h is the single most important expression in the course. It appears in the definition of the derivative (Chapter 2) and underpins nearly every computation that follows.
⚠️ Reading limits off a graph is a guaranteed exam question. Be precise about open vs. closed circles and about what happens from each side.
⚠️ Know how to determine whether a limit is +∞ or -∞ by analysing the sign of the numerator and the sign of a "small positive" or "small negative" denominator.
⚠️ One-sided limits and the existence theorem (two-sided limit exists iff both one-sided limits exist and agree) are tested frequently on their own and inside piecewise-function problems.
True or False: The slope of a secant line equals the instantaneous rate of change.
True or False: lim(x→a) f(x) can exist even if f(a) is undefined.
Fill in the blank: lim(x→a) f(x) = L if and only if lim(x→a⁻) f(x) = ___ and lim(x→a⁺) f(x) = ___.
True or False: If lim(x→3⁺) f(x) = ∞, we say the limit exists and equals infinity.
Fill in the blank: For a rational function g(x)/h(x), a vertical asymptote typically occurs where h(a) = 0 and g(a) ___ 0.
Answers: 1. False (it equals the average rate of change). 2. True. 3. L; L. 4. False (technically the limit does not exist; the notation describes the behaviour). 5. ≠ (is not equal to).
Q: Let f(x) = √(x + 1). Calculate the average rate of change on the interval [3, 8].
A: f(8) = √9 = 3, f(3) = √4 = 2. ARoC = (3 - 2) / (8 - 3) = 1/5.
Q: A rock on Mars has height h(t) = 10t - 2t². Find the average velocity on [1, 2] and on [0, 1], then estimate the instantaneous velocity at t = 1.
A: Average velocity on [1, 2] = [h(2) - h(1)] / 1 = [12 - 8] / 1 = 4... wait, h(2) = 20 - 8 = 12, h(1) = 10 - 2 = 8, so (12 - 8)/1 = 4. On [0, 1]: h(1) - h(0) = 8 - 0 = 8. The instantaneous velocity at t = 1 is approximately the average of these two: about 6 m/s (the exact value, via the limit, is 6 m/s).
Q: Given the piecewise function f(x) = { x² + 4 if x < 1; 0 if x = 1; 3 - x if x > 1 }, find lim(x→1) f(x).
A: Left-hand limit: lim(x→1⁻) (x² + 4) = 1 + 4 = 5. Right-hand limit: lim(x→1⁺) (3 - x) = 3 - 1 = 2. Since 5 ≠ 2, the two-sided limit does not exist (DNE).
Q: Find the vertical asymptotes of y = (x² + 2x + 1) / (x + x²).
A: Factor the denominator: x(x + 1). Set each factor to zero: x = 0 and x = -1. The numerator at x = 0 is 1 ≠ 0, and at x = -1 is 0 (so check further: (x + 1)² / x(x + 1) simplifies to (x + 1)/x, which has a hole at x = -1 and a vertical asymptote at x = 0 only).
Q: Evaluate lim(x→1⁺) (x² + 2x + 1) / (x + x²) by sign analysis.
A: After cancellation the function simplifies to (x + 1)/x near x = 0. As x → 0⁺, numerator → 1 (positive), denominator → 0⁺ (small positive). So the limit is +∞.
This material feeds directly into Section 6 (calculating limits algebraically using limit laws) and Section 8 (continuity and the Intermediate Value Theorem). The difference quotient introduced here becomes the formal definition of the derivative in Chapter 2, Section 1. Mastering one-sided limits now also pays off when you classify discontinuities later.
secant line, tangent line, average rate of change, ARoC, instantaneous rate of change, difference quotient, slope of tangent, limit, one-sided limit, left-hand limit, right-hand limit, two-sided limit, vertical asymptote, infinite limit, DNE, does not exist, limit does not exist, piecewise function limits, graphical limits, MTH 132, Calculus I, Chapter 1, Stewart Calculus, Section 1.4, Section 1.5