Difficulty: Foundational | Prerequisites: College Algebra (MAT 117) or Precalculus (MAT 170)
This is where Brief Calculus begins. Limits give you the language for describing what happens to a function's output as its input approaches a particular value, which underpins every major concept in the course. Average rate of change, built on that foundation, measures how quickly a quantity shifts over an interval and sets the stage for instantaneous rate of change (the derivative). If you are coming in cold, make sure you are comfortable with function notation, evaluating expressions, and basic graph reading.
A limit tells you the value a function is heading toward as the input gets close to some number, even if the function never arrives there. Average rate of change measures how much the output of a function changes per unit of input over a given interval, which is the slope of the straight line (secant line) connecting two points on a graph. Together, these two ideas build the bridge from algebra to calculus: the derivative is what you get when you shrink that interval down to nothing.
Limit
The value that f(x) approaches as x gets closer and closer to a particular number c. Written lim (x→c) f(x) = L. In simple terms, this means: "Where is the function heading?" You care about the journey, not whether it arrives.
One-sided limit
The value f(x) approaches when x comes from only one direction. lim (x→c⁻) is the left-hand limit; lim (x→c⁺) is the right-hand limit. Think of it as checking each side of a doorway separately.
Continuity
A function is continuous at x = c when three things hold: f(c) exists, lim (x→c) f(x) exists, and they are equal. In simple terms, you can draw through that point without lifting your pen.
Average rate of change
The ratio [f(b) - f(a)] / (b - a), measuring how much the output changes per unit of input over the interval [a, b]. Think of it as the slope between two points on the curve.
Secant line
The straight line connecting two points on a graph. Its slope equals the average rate of change between those two points.
Difference quotient
The expression [f(x + h) - f(x)] / h, which gives the average rate of change over an interval of width h starting at x. This is the formula that, once you take its limit as h→0, produces the derivative.
The notation lim (x→c) f(x) = L means: as x gets arbitrarily close to c (from both sides), f(x) gets arbitrarily close to L.
You never need f(c) itself to exist. The limit is about behaviour near c, not at c.
If plugging in c gives you a normal number (no division by zero, no 0/0), the limit equals f(c). This is called direct substitution.
Left-hand limit, lim (x→c⁻): approach c from values smaller than c.
Right-hand limit, lim (x→c⁺): approach c from values larger than c.
The two-sided limit exists only when both one-sided limits exist and are equal. If they disagree, the limit does not exist (DNE).
Example: for a piecewise function that equals x² when x < 2 and equals 5 when x ≥ 2, the left-hand limit at x = 2 is 4, the right-hand limit is 5, so the two-sided limit does not exist.
Direct substitution: try plugging in first. If it works, you are done.
Factoring and cancelling: when substitution gives 0/0, factor the numerator and denominator, cancel the common factor, then substitute again. Example: lim (x→3) (x² - 9)/(x - 3) = lim (x→3) (x + 3)(x - 3)/(x - 3) = lim (x→3) (x + 3) = 6.
Rationalising: when a square root creates 0/0, multiply numerator and denominator by the conjugate.
Limits at infinity: divide every term by the highest power of x in the denominator. Terms with x in the denominator vanish.
Continuous at c means: (1) f(c) is defined, (2) lim (x→c) f(x) exists, (3) lim (x→c) f(x) = f(c).
Polynomials, exponentials, and logarithms (on their domains) are continuous everywhere.
Rational functions are continuous everywhere except where the denominator is zero.
Discontinuities come in types: removable (a hole you could fill), jump (the two sides disagree), and infinite (a vertical asymptote).
Formula: AROC = [f(b) - f(a)] / (b - a).
Geometrically, this is the slope of the secant line through (a, f(a)) and (b, f(b)).
Units: the units of f divided by the units of x. If f is revenue in dollars and x is time in months, the AROC is dollars per month.
Example: if f(x) = x² and you want the average rate of change from x = 1 to x = 4, then AROC = (16 - 1)/(4 - 1) = 15/3 = 5.
Written as [f(x + h) - f(x)] / h.
This is a generalised average rate of change over an interval from x to x + h.
Simplifying the difference quotient for a given function is a core skill. You expand f(x + h), subtract f(x), then divide by h and cancel.
Example: for f(x) = 3x², the difference quotient is [3(x + h)² - 3x²] / h = [3x² + 6xh + 3h² - 3x²] / h = (6xh + 3h²) / h = 6x + 3h.
\lim_{x \to c} f(x) = L\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}\text{Difference Quotient} = \frac{f(x+h) - f(x)}{h}Key limit laws (assuming both limits exist):
Sum law: lim [f(x) + g(x)] = lim f(x) + lim g(x)
Product law: lim [f(x) · g(x)] = lim f(x) · lim g(x)
Quotient law: lim [f(x) / g(x)] = lim f(x) / lim g(x), provided lim g(x) ≠ 0
Constant multiple law: lim [k · f(x)] = k · lim f(x)
Power law: lim [f(x)]ⁿ = [lim f(x)]ⁿ
Average rate of change is what you calculate every time you divide "how much something changed" by "how long it took." Average speed on a road trip (miles divided by hours), average monthly revenue growth, or the average rate a drug concentration falls in the bloodstream are all average rates of change.
Limits show up whenever you want to know a trend's destination. An economist asking "what happens to cost per unit as production grows very large?" is evaluating a limit at infinity.
Students often think that if f(c) is undefined, then lim (x→c) f(x) does not exist. That is wrong. The limit can exist perfectly well even when the function has a hole at c.
Students sometimes assume that "limit does not exist" means "the function blows up to infinity." In fact, a limit can fail to exist simply because the left-hand and right-hand limits disagree (a jump), with no infinity involved.
Confusing the average rate of change with the instantaneous rate of change. The average rate of change uses two points; the instantaneous rate uses one point and a limiting process.
Forgetting to check both sides when a piecewise function is involved. You need the left-hand and right-hand limits to agree.
⚠️ The difference quotient appears on the midterm. Be able to simplify it for polynomial and simple rational functions from memory.
⚠️ Know all three conditions for continuity. Exam questions love giving you a piecewise function and asking you to find a value of a parameter that makes it continuous.
⚠️ Factoring to resolve 0/0 indeterminate forms is tested repeatedly. Practise until it is automatic.
True or false: if f(2) = 7, then lim (x→2) f(x) = 7. Answer: False. The function value and the limit can differ (or the limit may not exist).
Fill in the blank: the average rate of change of f(x) from x = a to x = b equals the slope of the ______ line. Answer: Secant.
True or false: if the left-hand limit at c equals 4 and the right-hand limit at c equals 4, the two-sided limit equals 4. Answer: True.
Fill in the blank: the difference quotient is [f(x + h) - f(x)] divided by ______. Answer: h.
Q: Evaluate lim (x→2) (x² - 4) / (x - 2).
A: Factor the numerator: (x - 2)(x + 2) / (x - 2). Cancel (x - 2). You get lim (x→2) (x + 2) = 4.
Q: Find the average rate of change of f(x) = 3x² + 1 from x = 1 to x = 3.
A: f(1) = 4, f(3) = 28. AROC = (28 - 4) / (3 - 1) = 24 / 2 = 12.
Q: Simplify the difference quotient for f(x) = x² + 5x.
A: f(x + h) = (x + h)² + 5(x + h) = x² + 2xh + h² + 5x + 5h. Subtract f(x): 2xh + h² + 5h. Divide by h: 2x + h + 5.
Q: A function g is defined as g(x) = 2x + 1 when x < 3, and g(x) = x² - 2 when x ≥ 3. Is g continuous at x = 3?
A: Left-hand limit = 2(3) + 1 = 7. Right-hand limit = 9 - 2 = 7. g(3) = 7. All three match, so yes, g is continuous at x = 3.
Q: What is lim (x→∞) (5x² + 3) / (2x² - 1)?
A: Divide every term by x². You get (5 + 3/x²) / (2 - 1/x²). As x→∞, the fractions vanish. The limit is 5/2.
The difference quotient leads directly into the definition of the derivative (next topic). When you take lim (h→0) of the difference quotient, you get the instantaneous rate of change, which is the central object of differential calculus.
Continuity matters again when you reach the definite integral: the Fundamental Theorem of Calculus requires the integrand to be continuous on the interval of integration.
limit, limit of a function, evaluating limits, limit laws, one-sided limit, left-hand limit, right-hand limit, continuity, continuous function, removable discontinuity, jump discontinuity, average rate of change, AROC, secant line, slope of secant, difference quotient, indeterminate form 0/0, factoring limits, direct substitution, limits at infinity, MAT 210, Brief Calculus, ASU