Difficulty: Introductory | Prerequisites: basic algebra (factoring polynomials, simplifying rational expressions)
Limits are the foundation of calculus, the concept that every derivative and integral rests on. In MAT 210 (Brief Calculus) at ASU, limits appear right away in Quiz 1 and stay relevant through every later topic. You should already be comfortable factoring quadratics and simplifying rational expressions before tackling this material. If you can cancel common factors in a fraction, you have the algebra you need.
A limit describes what value a function approaches as the input gets close to some number (or heads to infinity). You find limits algebraically by factoring and cancelling, or by comparing polynomial degrees. You find them graphically by reading where the curve is heading from the left and right sides.
Limit
The value that f(x) approaches as x approaches a particular number c, written lim x→c f(x). Think of it as: "Where is the y-value heading as x gets closer and closer to c?"
One-sided limit (left-hand limit)
lim x→c⁻ f(x). The value f(x) approaches as x comes from values less than c (from the left on the number line). In simple terms, this means you only care about what happens on the left side of c.
One-sided limit (right-hand limit)
lim x→c⁺ f(x). The value f(x) approaches as x comes from values greater than c (from the right). In simple terms, you are only watching the curve from the right side.
Two-sided limit
lim x→c f(x) exists only when the left-hand and right-hand limits are equal. Think of it as: both sides must agree, or the limit does not exist (DNE).
DNE (Does Not Exist)
Written when a limit fails to settle on a single finite value. This happens when the left and right limits disagree, or the function blows up to infinity without settling.
Limit at infinity
lim x→∞ f(x). What the function approaches as x grows without bound. In simple terms, this is the long-run behaviour of the function as x gets very large.
Indeterminate form (0/0)
When direct substitution gives 0/0, the limit is not automatically zero or undefined. It means more work is needed (typically factoring and cancelling). Think of it as a "try again" signal, not an answer.
The first thing to try: plug the target value of x straight into the function.
If you get a real number, that is the limit. Done.
If you get 0/0, you have an indeterminate form and need another technique.
When direct substitution yields 0/0, factor the numerator and denominator, cancel the common factor, then substitute again.
Worked example from the quiz:
lim x→−1 of (x² + 14x + 13) / (x² + x)
Direct substitution gives (1 − 14 + 13) / (1 − 1) = 0/0. Indeterminate.
Factor the numerator: x² + 14x + 13 = (x + 1)(x + 13).
Factor the denominator: x² + x = x(x + 1).
Cancel the common factor (x + 1): you are left with (x + 13) / x.
Substitute x = −1: (−1 + 13) / (−1) = 12 / (−1) = −12.
Answer: −12.
When x → ∞, the behaviour of a rational function depends on the highest-power terms in the numerator and denominator.
Degree of numerator < degree of denominator: the limit is 0. The denominator grows faster and overwhelms the numerator.
Degree of numerator = degree of denominator: the limit is the ratio of the leading coefficients.
Degree of numerator > degree of denominator: the limit is ±∞ (or DNE, depending on convention).
Worked example from the quiz:
lim x→∞ of (−2x² + 14x + 13) / (x⁵ + x)
Numerator degree: 2. Denominator degree: 5.
Since 2 < 5, the denominator grows much faster.
Answer: 0.
To find a one-sided limit from a graph, trace the curve toward the target x-value from one direction only.
Right-hand limit (x → c⁺): follow the curve from the right. Read the y-value the curve is heading toward.
Left-hand limit (x → c⁻): follow the curve from the left. Read the y-value the curve is heading toward.
Open circles (hollow dots) on the graph mean the function is not defined at that point, but the curve still approaches a y-value there. The limit cares about approach, not arrival.
The two-sided limit lim x→c f(x) exists only if the left-hand and right-hand limits are equal. If they match, that shared value is the limit. If they disagree, write DNE.
The value f(c) and the limit lim x→c f(x) are different questions.
The limit asks: "Where is the curve heading?"
The function value asks: "Where is the dot actually plotted?"
A solid dot at a different y-value than where the curve approaches means f(c) differs from the limit.
Worked example from the quiz (at x = −8):
lim x→−8⁺ f(x) = −6 (approaching from the right, the curve heads toward y = −6)
lim x→−8⁻ f(x) = −6 (approaching from the left, same destination)
lim x→−8 f(x) = −6 (both sides agree)
f(−8) = −3 (the solid dot sits at y = −3, not at −6)
The limit and the function value are not the same here. This is a classic exam scenario.
Direct substitution rule: If f is continuous at c, then lim x→c f(x) = f(c).
Factoring technique: When substitution gives 0/0, factor both numerator and denominator, cancel the shared factor, then substitute again.
Degree comparison for limits at infinity (rational functions p(x)/q(x)):
deg(p) < deg(q) → limit = 0
deg(p) = deg(q) → limit = leading coefficient of p / leading coefficient of q
deg(p) > deg(q) → limit = ±∞ (DNE in many course conventions)
Two-sided limit existence rule: lim x→c f(x) exists if and only if lim x→c⁻ f(x) = lim x→c⁺ f(x).
Limits are how engineers and scientists describe behaviour at boundaries. When a pharmacologist models drug concentration in the bloodstream over time, the limit as t → ∞ tells them whether the drug fully clears or plateaus at a residual level. In economics, a limit at infinity on a cost function reveals whether average cost per unit stabilises as production scales up.
Students often think the limit and the function value must be the same. They do not. A function can have a limit of −6 at a point where the actual value is −3 (as in the quiz's graph question).
Students often think 0/0 means the limit is zero or undefined. It does not. 0/0 is an indeterminate form, which means you need to do more algebra before you have an answer.
Students often confuse "limit at infinity equals 0" with "the function never touches zero." The limit describes the trend, not whether the function literally reaches that value.
Students sometimes read a graph's open circle as meaning the limit does not exist. The open circle means the function is not defined there, but the limit can still exist if the curve approaches a specific y-value from both sides.
⚠️ Expect at least one problem where direct substitution gives 0/0 and you must factor to find the limit. This is the most common algebraic limit question in MAT 210.
⚠️ Graph-based limit questions will test whether you can distinguish between f(c) and lim x→c f(x). If these are different on the graph, the exam wants you to show you know the difference.
⚠️ Limits at infinity with rational functions come down to comparing degrees. Know the three cases cold: numerator degree less than, equal to, or greater than the denominator degree.
⚠️ One-sided limits: if a question asks for both lim x→c⁺ and lim x→c⁻, it is setting up a follow-up about whether the two-sided limit exists.
True or false: If f(3) = 5, then lim x→3 f(x) must also equal 5. False. The function value and the limit can differ.
Fill in the blank: When direct substitution gives 0/0, this is called a(n) __________ form. Indeterminate.
True or false: If the degree of the numerator is less than the degree of the denominator, the limit as x → ∞ is 0. True.
Fill in the blank: The two-sided limit exists only when the __________ limit equals the __________ limit. Left-hand, right-hand.
True or false: An open circle on a graph means the limit does not exist at that point. False. It means the function is not defined there, but the limit may still exist.
Q: Find lim x→−1 of (x² + 14x + 13) / (x² + x). Show your steps.
A: Factor numerator as (x + 1)(x + 13) and denominator as x(x + 1). Cancel (x + 1). Substitute x = −1 into (x + 13)/x to get 12/(−1) = −12.
Q: Find lim x→∞ of (−2x² + 14x + 13) / (x⁵ + x).
A: The numerator has degree 2 and the denominator has degree 5. Since 2 < 5, the limit is 0.
Q: On a graph, f(−8) = −3 but the curve approaches y = −6 from both sides. What is lim x→−8 f(x)?
A: The limit is −6. The limit depends on where the curve approaches, not on the function value.
Q: If lim x→c⁺ f(x) = 4 and lim x→c⁻ f(x) = 7, what is lim x→c f(x)?
A: DNE. The one-sided limits are not equal, so the two-sided limit does not exist.
Q: Direct substitution into a limit gives 0/0. What should you do next?
A: Factor the numerator and denominator, cancel the common factor, and try substitution again.
Limits lead directly into derivatives: the derivative is itself defined as a limit (lim h→0 of [f(x+h) − f(x)] / h). Every rule you learn here, especially factoring and cancelling indeterminate forms, reappears when you compute derivatives from the definition. Limits at infinity also connect to horizontal asymptotes, which come up when you sketch rational functions.
Limits, algebraic limits, graphical limits, one-sided limit, left-hand limit, right-hand limit, two-sided limit, limit at infinity, indeterminate form, 0/0, factoring limits, polynomial degree comparison, DNE, does not exist, open circle, function value vs limit, continuity, MAT 210, Brief Calculus, ASU, Arizona State University, Quiz 1