Source: Chapter 10.1 Lecture Notes | Brief Calculus
Tags: functions, limits, left-hand limit, right-hand limit, independent variable, dependent variable, DNE, does not exist, calculus, MAT 210, brief calculus
A function is a rule that assigns exactly one output to every input, written as y = f(x). A limit describes what value a function's output approaches as the input gets closer and closer to some target value, without necessarily reaching it. Limits exist only when the left-hand and right-hand approaches agree; if they don't, the limit does not exist (DNE).
Difficulty: Introductory | Prerequisites: Basic algebra (substitution, evaluating expressions, simplifying fractions).
Chapter 10.1 is where calculus properly begins. Everything before this, algebraic manipulation, graphing, domain and range, was setup. Functions formalise the idea that one quantity depends on another. Limits then let you ask: "What happens to the output as the input gets arbitrarily close to some value?" This question is the foundation of derivatives and integrals, which are the two main pillars of calculus. If you are comfortable substituting values into expressions and simplifying rational expressions, you have what you need to follow this material.
Function
A rule that assigns exactly one output for every input. In simple terms, you put a number in, the rule does something to it, and you get exactly one number out.
Independent variable (x)
The input value you control or choose. Think of it as the dial you turn.
Dependent variable (y)
The output value that depends on what you put in for x. It is determined by the function rule.
Function notation: y = f(x)
The standard way of writing "y is a function of x." f(x) does not mean f times x. It means "the output of the function f when the input is x."
Limit
The value that f(x) approaches as x gets closer and closer to a particular number. In simple terms, it is the answer to "Where is this heading?" rather than "Where is it right now?"
Left-hand limit
The value f(x) approaches as x approaches a target from values less than the target (from the left on a number line). Written with a superscript minus: lim x→a⁻ f(x).
Right-hand limit
The value f(x) approaches as x approaches a target from values greater than the target (from the right on a number line). Written with a superscript plus: lim x→a⁺ f(x).
Overall (two-sided) limit
The single value L such that both the left-hand and right-hand limits equal L. Written as lim x→a f(x) = L. This exists only when both one-sided limits agree.
DNE (Does Not Exist)
Used when the left-hand and right-hand limits are not equal, or when the function's behaviour near the target value does not settle toward a single number. Think of it as: the function can't make up its mind.
Domain
The set of all input values for which the function is defined. If plugging in a value causes division by zero or another undefined operation, that value is not in the domain.
What makes something a function
Each input (x) maps to exactly one output (y). One input, one answer, no exceptions.
If a single x value could produce two different y values, the rule is not a function.
Notation and evaluation
y = f(x) is read "y equals f of x." The letter f names the rule; x is the input you supply.
To evaluate f(3) when f(x) = x² + 1: replace every x with 3, then compute. f(3) = 3² + 1 = 9 + 1 = 10.
The result (10 in this case) is called the output or the value of the function at x = 3.
Independent vs dependent
x is independent: you pick it.
y is dependent: the function determines it once x is chosen.
On a graph, x runs along the horizontal axis and y along the vertical.
The central idea
A limit asks: "As x gets closer and closer to some value a, what does f(x) get closer to?"
You are not asking what f(a) equals. You are asking what the outputs are heading toward. The function does not even need to be defined at a for the limit to exist.
Building intuition with a table
Consider f(x) = 2x + 1 as x approaches 4.
From the left: x = 3.9 → 8.8, x = 3.99 → 8.98, x = 3.999 → 8.998, x = 3.9999 → 8.9998. The outputs are creeping toward 9.
From the right: x = 4.1 → 9.2, x = 4.01 → 9.02, x = 4.001 → 9.002, x = 4.0001 → 9.0002. Also heading toward 9.
Both sides agree, so lim x→4 f(x) = 9.
Left-hand and right-hand limits
Left-hand limit (x → a⁻): approach a from values smaller than a.
Right-hand limit (x → a⁺): approach a from values larger than a.
The superscript minus or plus tells you the direction. Think of the number line: minus means coming from the left, plus from the right.
When does the overall limit exist?
The two-sided limit lim x→a f(x) = L exists if and only if:
The left-hand limit equals L, and
The right-hand limit equals L.
If the two one-sided limits give different values, the overall limit does not exist (DNE).
Limits where the function is undefined
Example: f(x) = (x² - 4) / (x - 2). At x = 2, you get 0/0, so x = 2 is not in the domain.
But the limit still makes sense. Factor the numerator: (x² - 4) = (x - 2)(x + 2). Cancel (x - 2) to get x + 2.
As x → 2, x + 2 → 4. So lim x→2 (x² - 4)/(x - 2) = 4, even though f(2) itself is undefined.
Graphical clues: open and filled circles
An open circle on a graph at a point means the function is not defined there, but the limit can still exist (the curve approaches that point from both sides).
A filled circle means the function is defined and equals that value. If the filled circle sits away from where the curve is heading, the limit is about the curve's direction, not the dot's position.
Function notation
y = f(x), where x is input and y is output.
Evaluating a function
Substitute the input value for every x in the expression. Example: if f(x) = x² + 1, then f(3) = 3² + 1 = 10.
Limit notation
lim x→a f(x) = L means "as x approaches a, f(x) approaches L."
One-sided limit notation
lim x→a⁻ f(x): left-hand limit (approaching from below)
lim x→a⁺ f(x): right-hand limit (approaching from above)
Factoring trick for 0/0 limits
When direct substitution gives 0/0, factor the numerator and denominator, cancel the common factor, then substitute again. Example: lim x→2 (x² - 4)/(x - 2) = lim x→2 (x + 2) = 4.
Limits are behind every rate-of-change calculation in engineering, economics and the sciences. When an economist asks "What happens to revenue as we push the price toward £50?", that is a limit. When a physicist calculates instantaneous velocity (speed at a single moment rather than an average over a time interval), limits are the tool that makes that precise. The derivative, which you will meet next, is defined as a limit.
"The limit is the same as f(a)." It is not. The limit describes where f(x) is heading as x approaches a. The function might not even be defined at a, yet the limit can still exist. The (x² - 4)/(x - 2) example shows exactly this.
"If the function is undefined at a point, the limit can't exist there." Wrong. The limit cares about the neighbourhood around the point, not the point itself.
"f(x) notation means f multiplied by x." A common early mistake. f(x) names a function evaluated at x, not a product.
"Open circle on a graph means the limit doesn't exist." The opposite. An open circle typically means the function isn't defined there, but the curve still approaches a value, so the limit does exist. A filled circle sitting away from the curve's path is the signal that the function value and the limit disagree.
⚠️ Expect to evaluate limits by building a table of values approaching from both sides. Know the procedure cold.
⚠️ Be ready to factor and cancel when direct substitution gives 0/0. The (x² - 4)/(x - 2) type problem is a staple.
⚠️ You will be asked whether a limit exists given a graph. Check both one-sided limits. If they differ, write DNE.
⚠️ Know the notation: the superscript minus and plus on one-sided limits, and when to use each.
⚠️ Distinguish between the value of f(a) and the value of lim x→a f(x). Exam questions test this directly.
True or false: f(x) notation means "f multiplied by x."
False. It means "the function f evaluated at input x."
Fill in the blank: The limit of f(x) as x → a exists only when the ______ limit and the ______ limit are equal.
left-hand, right-hand
True or false: If f(2) is undefined, then lim x→2 f(x) must also be undefined.
False. The limit can exist even when the function is not defined at that point.
Fill in the blank: DNE stands for ______.
Does Not Exist
True or false: On a graph, an open circle at a point means the limit does not exist.
False. An open circle means the function is not defined there, but the limit may well exist.
Q: If f(x) = x² + 1, what is f(5)?
A: f(5) = 5² + 1 = 25 + 1 = 26.
Q: Using a table of values, estimate lim x→3 (2x + 1).
A: As x approaches 3 from both sides, f(x) approaches 7. Try x = 2.9 (6.8), x = 2.99 (6.98), x = 3.01 (7.02), x = 3.1 (7.2). The outputs converge on 7, so the limit is 7.
Q: Evaluate lim x→2 (x² - 4)/(x - 2).
A: Direct substitution gives 0/0. Factor: (x² - 4) = (x - 2)(x + 2). Cancel (x - 2) to get x + 2. Then lim x→2 (x + 2) = 4.
Q: Suppose the left-hand limit of g(x) as x → 5 is 3, and the right-hand limit is 7. Does lim x→5 g(x) exist?
A: No. The one-sided limits are not equal (3 ≠ 7), so the overall limit does not exist (DNE).
Q: On a graph, you see an open circle at the point (4, 6) and the curve approaches that point from both sides. What is lim x→4 f(x)?
A: The limit is 6. The open circle means f(4) is not defined, but both sides of the curve head toward y = 6, so the limit exists and equals 6.
This connects directly to Chapter 10.2 and beyond, where limits are used to define the derivative. The derivative measures instantaneous rate of change, and it is built on the limit of a difference quotient. If you understand what a limit does, the definition of the derivative will make sense rather than feel like a formula dropped from nowhere.
Continuity (whether a function has any gaps or jumps) is also defined using limits: a function is continuous at a point when the limit equals the function's value there. That idea will recur throughout the course.
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