Source: MTH 132 Chapter 1, Michigan State University
Difficulty: Introductory | Prerequisites: Solid algebra (factoring, rationalising, absolute value), basic trigonometry (unit circle values, sin, cos, tan), familiarity with piecewise functions.
Chapter 1 is the gateway to all of calculus. It builds two ideas that everything else rests on: how fast a quantity is changing (rates of change) and what value a function approaches (limits). You will need these concepts for derivatives in Chapter 2 and integrals later on. If you are comfortable with slopes from algebra and basic function evaluation, you have enough to start here.
Slope between two points on a curve gives you the average rate of change (secant line); shrinking that gap to a single point gives the instantaneous rate of change (tangent line), and the tool that makes that shrinking precise is the limit. Limits can be evaluated graphically, numerically, or algebraically, and when a function blows up near a point you get a vertical asymptote instead of a finite answer.
Secant line
A line that passes through two distinct points on the graph of a function. Its slope equals the average rate of change between those two points. In simple terms, it is the straight line you draw between two spots on a curve to measure the overall steepness between them.
Tangent line
A line that touches the curve at exactly one point and has the same direction as the curve at that point. Its slope equals the instantaneous rate of change. Think of it as what the secant line becomes when the two points are squeezed together until they merge into one.
Average rate of change (ARoC)
The ratio of the change in the function's output to the change in its input over an interval: (f(x₂) − f(x₁)) / (x₂ − x₁). Units are always "units of f per unit of x." In simple terms, this is the slope of the secant line.
Instantaneous rate of change (IRoC)
The rate at which a function is changing at a single point, obtained by taking the limit of the average rate of change as the interval shrinks to zero. Units are the same as ARoC. In simple terms, this is the slope of the tangent line, or the speedometer reading at one instant.
Limit
We write lim (x→a) f(x) = L to mean that the values of f(x) can be made as close to L as we like by choosing x sufficiently close to a (but not equal to a). The function does not need to equal L at a, or even be defined there. Think of it as asking "what value is f(x) heading towards as x gets really close to a?"
Left-hand limit
lim (x→a⁻) f(x) = L means f(x) approaches L as x approaches a from values less than a (from the left on the number line).
Right-hand limit
lim (x→a⁺) f(x) = L means f(x) approaches L as x approaches a from values greater than a (from the right on the number line).
Two-sided limit existence rule
lim (x→a) f(x) = L exists if and only if both the left-hand limit and the right-hand limit exist and are equal to each other.
Vertical asymptote
The vertical line x = a is a vertical asymptote of y = f(x) if at least one of the one-sided limits at a is +∞ or −∞. This typically happens when the denominator of a fraction goes to zero while the numerator does not. Think of it as a wall the graph shoots up (or down) along but never crosses.
Infinite limit
A limit where f(x) grows without bound (positively or negatively) as x approaches a. We write lim (x→a) f(x) = ∞ or −∞. Technically these limits "do not exist" as finite numbers, but the notation tells us the specific way they fail.
A secant line requires two ingredients: a function and two specified points on it.
The slope of the secant line through P(x₁, f(x₁)) and Q(x₂, f(x₂)) is: m = (f(x₂) − f(x₁)) / (x₂ − x₁).
This slope is identical to the average rate of change of f on the interval [x₁, x₂].
Units of the average rate of change are always (units of the output) / (units of the input). For example, if distance is in miles and time is in minutes, the ARoC is in miles per minute.
Identify the two x-values (or input values) and compute the corresponding f-values.
Subtract: Δf = f(x₂) − f(x₁) and Δx = x₂ − x₁.
Divide: ARoC = Δf / Δx.
Attach units and interpret the result in context.
Example: f(x) = √(x + 1) on [3, 8]. Then f(3) = 2 and f(8) = 3, so the ARoC = (3 − 2)/(8 − 3) = 1/5.
A tangent line requires a function and one specified point.
It just barely skims the graph at that point, matching the curve's direction there.
You can think of the tangent line as the secant line in the limit: slide Q along the curve towards P, and the secant line rotates into the tangent line.
The slope of the tangent line at a point is the instantaneous rate of change at that point.
To estimate the tangent slope without calculus, compute the secant slope through points very close to the target on either side, then average or observe the trend.
For f(x) = x² − 3 at x = 2:
Pick a nearby interval, say [2, 2.001], and compute the secant slope.
Pick another, say [1.999, 2], and compute that secant slope.
As the interval shrinks, the secant slopes converge to the tangent slope (which turns out to be 4).
If h(t) gives position at time t, then the average velocity over [t₁, t₂] = (h(t₂) − h(t₁)) / (t₂ − t₁). This is the slope of the secant line on the position graph.
The instantaneous velocity at time t = a is the slope of the tangent line to h(t) at t = a.
Units of both average and instantaneous rate of change are (units of f) / (units of x), e.g. metres per second.
lim (x→a) f(x) = L means: we can force f(x) as close to L as we wish by choosing x sufficiently close to a (but not equal to a).
The value of f(a) itself is irrelevant to the limit. The function can be undefined at a, or defined to something different from L, and the limit can still be L.
Left-hand limit (x→a⁻): x approaches a from values below a.
Right-hand limit (x→a⁺): x approaches a from values above a.
The two-sided limit exists and equals L if and only if both one-sided limits exist and both equal L.
Trace the curve from the left towards x = a; whatever y-value you approach is the left-hand limit.
Trace from the right; that y-value is the right-hand limit.
If the two agree, the two-sided limit exists and equals that value.
Open circles on the graph indicate the function is not defined there (or is defined elsewhere). Limits care about approach, not arrival.
To find lim (x→a⁻) f(x), use whichever piece of the function applies for x just below a.
To find lim (x→a⁺) f(x), use the piece that applies for x just above a.
Compare the two results. If they match, that is the two-sided limit. If they differ, the two-sided limit does not exist.
Example: f(x) = { x² + 4 if x < 1, 0 if x = 1, 3 − x if x > 1 }.
lim (x→1⁻) f(x) = 1² + 4 = 5
lim (x→1⁺) f(x) = 3 − 1 = 2
Since 5 ≠ 2, lim (x→1) f(x) does not exist.
Note: f(1) = 0, but that plays no role in the limit.
When f(x) grows without bound (positively or negatively) as x→a, we say the limit is ∞ or −∞.
The line x = a is a vertical asymptote if at least one of these holds: lim (x→a⁺) f(x) = ±∞, lim (x→a⁻) f(x) = ±∞, or lim (x→a) f(x) = ±∞.
Quick detection for rational functions g(x)/h(x): if g(a) ≠ 0 and h(a) = 0, there is very likely a vertical asymptote at x = a. Confirm by checking the sign of each one-sided limit.
Plug in a to confirm you are in the N/0 case (numerator non-zero, denominator zero).
Determine the sign of the numerator at a.
Check the sign of the denominator from the left and from the right (use a test value very close to a on each side).
Positive/positive or negative/negative → +∞. Mixed signs → −∞.
Example: lim (x→1⁺) 5/(1 − x). Numerator at 1 is 5 (positive). Denominator 1 − x is negative when x is just above 1. Positive / negative → −∞.
Slope of the secant line (average rate of change)
m = (f(x₂) − f(x₁)) / (x₂ − x₁)
Slope of the tangent line (instantaneous rate of change)
m = lim (h→0) [f(a + h) − f(a)] / h
(This is a preview of the derivative definition coming in Chapter 2.)
Equation of the tangent line at x = a
y − f(a) = m(x − a), where m is the slope of the tangent line at x = a.
Average and instantaneous rates of change appear everywhere outside pure mathematics. Velocity is the most familiar example (rate of change of position), but the same idea drives how we measure the infection rate of a disease, the rate of a chemical reaction, stock price movement, population growth, acceleration (rate of change of velocity), monthly sales trends, and temperature change over time. Any time you want to know "how fast is this quantity changing," you are computing a rate of change.
Students often think the limit of f(x) as x→a must equal f(a). It does not. The function can be undefined at a, or defined to a completely different value, and the limit can still exist.
Students confuse "the limit is ∞" with "the limit exists." When a limit is ∞ or −∞, it technically does not exist as a finite number. The infinity notation describes the behaviour, but the answer to "does the limit exist?" is no.
Students sometimes believe a tangent line can touch the graph at only one point total. In fact, a tangent line can cross the curve elsewhere; what matters is that it matches the curve's direction at the one point of tangency.
Students forget to check both one-sided limits before claiming a two-sided limit exists. If the left and right limits disagree, the two-sided limit does not exist, regardless of what f(a) equals.
⚠️ Computing the average rate of change from a table or a formula is a near-certain exam question. Always include units in your answer.
⚠️ Be prepared to evaluate one-sided and two-sided limits from a graph with open and closed circles. Know what each circle type means.
⚠️ Piecewise-function limit problems appear frequently. The procedure is always the same: evaluate the left-hand limit using the piece for x < a, the right-hand limit using the piece for x > a, compare.
⚠️ Vertical asymptote detection (the N/0 case) is a classic exam topic. You must check the sign on each side to determine whether the limit is +∞ or −∞.
⚠️ Estimating the slope of a tangent line by taking secant slopes through increasingly close points is tested both as a concept question and as a computation.
True or False: If f(3) = 7, then lim (x→3) f(x) must equal 7.
The slope of the secant line through (2, f(2)) and (5, f(5)) gives the __________ rate of change on [2, 5].
True or False: The line x = 2 is a vertical asymptote of f(x) = (x + 1)/(x − 2).
If lim (x→4⁻) f(x) = 6 and lim (x→4⁺) f(x) = 6, then lim (x→4) f(x) = __________.
True or False: A tangent line at a point can never cross the curve at any other point.
Answers: 1. False (the limit depends on what f(x) approaches, not on f(3)). 2. average. 3. True (numerator is 3 ≠ 0 when x = 2, denominator is 0). 4. 6. 5. False (it can cross the curve elsewhere; it only matches the curve's direction at the point of tangency).
Q: A rock thrown upward on Mars has height h(t) = 10t − 2t². Find the average velocity over [1, 2] and over [0, 1], then estimate the instantaneous velocity at t = 1.
A: Average velocity on [1, 2]: h(2) = 12, h(1) = 8, so (12 − 8)/(2 − 1) = 4 m/s. Average velocity on [0, 1]: h(1) = 8, h(0) = 0, so (8 − 0)/(1 − 0) = 8 m/s. The instantaneous velocity at t = 1 is approximately the average of these two: (4 + 8)/2 = 6 m/s. (The exact answer, via the derivative, is also 6 m/s.)
Q: Let f(x) = { x² + 4 if x < 1, 0 if x = 1, 3 − x if 1 < x ≤ 2, x/2 if x > 2 }. Evaluate lim (x→1) f(x) and lim (x→2) f(x).
A: For x→1: left-hand limit uses x² + 4, giving 5. Right-hand limit uses 3 − x, giving 2. Since 5 ≠ 2, the two-sided limit does not exist. For x→2: left-hand limit uses 3 − x, giving 1. Right-hand limit uses x/2, giving 1. Both sides agree, so lim (x→2) f(x) = 1.
Q: Find the vertical asymptotes of y = (x² + 2x + 1)/(x + x²).
A: Factor: numerator = (x + 1)², denominator = x(x + 1). Cancel (x + 1) to get (x + 1)/x, valid when x ≠ −1. The denominator x = 0 gives a vertical asymptote at x = 0 (numerator is 1 ≠ 0 there). At x = −1, the original function is undefined but the simplified form gives 0/( −1) = 0, so x = −1 is a removable discontinuity (a hole), not a vertical asymptote.
Q: Evaluate lim (x→1⁺) 5/(1 − x).
A: When x is just above 1, the denominator 1 − x is a small negative number, and the numerator is 5 (positive). Positive divided by small negative → −∞.
The tangent-line slope you approximate here is exactly the derivative you will define formally in Chapter 2. Everything in Section 4 is motivation for the derivative formula. The limit concept from Section 5 is the engine behind derivatives, integrals, and series for the rest of calculus. Vertical asymptotes reappear when you study curve sketching and improper integrals.
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