Difficulty: Introductory | Prerequisites: Basic algebra (factoring, fractions, square roots)
Limits are the foundation of everything in calculus. Before you can talk about derivatives, rates of change, or anything else in this course, you need to understand what it means for a function to "approach" a value. This chapter introduces the concept and the core techniques for evaluating limits, especially what to do when plugging in directly gives you something undefined. If you are comfortable simplifying fractions and factoring polynomials, you have what you need to start here.
A limit tells you where a function's output is headed as the input approaches some target value, even if the function is not defined at that exact point. When direct substitution gives a real number, that is the limit. When it gives 0/0 (an indeterminate form), you need another technique: a table of values, factoring, or multiplying by a conjugate.
Tags: limit, limit of a function, approaching a value, limit notation, lim
Limit
The value that f(x) approaches as x gets closer and closer to some number c, from both sides. Written as lim (x→c) f(x) = L.
Think of it as: the destination the output is heading towards, even if it never arrives.
Indeterminate form
The result 0/0 when you substitute the target value into a function. It does not mean the limit is zero or undefined. It means substitution alone cannot tell you the answer and you need a different method.
In simple terms, 0/0 is a "try again" signal, not an answer.
One-sided limit
The value f(x) approaches as x comes from only the left (x < c) or only the right (x > c). The full (two-sided) limit exists only when both one-sided limits agree.
Think of it as: checking both directions of traffic before confirming the road is clear.
Conjugate
The expression you get by flipping the sign between two terms involving a square root. The conjugate of (√x − 1) is (√x + 1). Multiplying a radical expression by its conjugate eliminates the root from the numerator or denominator.
In simple terms, it is the algebraic trick that clears square roots out of fractions.
Continuous function
A function where f(c) equals lim (x→c) f(x), meaning there are no gaps, jumps, or holes at that point. If a function is continuous at c, you can find the limit by direct substitution.
Think of it as: a graph you can draw without lifting your pencil.
Tags: evaluating limits, direct substitution, table of values, limit techniques, rationalizing
Plug the target value straight into the function.
If you get a real number, that number is the limit. Done.
Works whenever the function is continuous at the target point.
Example: f(x) = x² + 5, find lim (x→0).
f(0) = 0² + 5 = 5.
No issues, so the limit is 5.
Used when substitution gives 0/0 and you want numerical confirmation.
Choose x-values approaching the target from both sides.
Compute f(x) for each and watch where the outputs converge.
Example: S(t) = (−2t³ + 19t² − 8t − 9) / (−t² + 8t − 7), find lim (t→1).
S(1) gives 0/0, so build a table:
t = 0.95 → 3.859; t = 0.98 → 3.943; t = 0.9 → 3.497
t = 1.001 → 4.003; t = 1.01 → 4.028; t = 1.1 → 4.385
From both sides, the outputs close in on 4.
Therefore lim (t→1) S(t) = 4.
Used when the 0/0 form involves a square root.
Multiply the numerator and denominator by the conjugate of the radical expression.
This eliminates the root and reveals a factor that cancels with the denominator.
Example: find lim (x→1) (√x − 1) / (x − 1).
Direct substitution gives (1 − 1) / (1 − 1) = 0/0.
Multiply top and bottom by (√x + 1):
Numerator becomes (√x − 1)(√x + 1) = x − 1.
Expression simplifies to (x − 1) / [(x − 1)(√x + 1)] = 1 / (√x + 1).
Now substitute: 1 / (√1 + 1) = 1/2.
Step 1: Try direct substitution.
Step 2: If you get a number, stop. That is the limit.
Step 3: If you get 0/0, check for a factorable polynomial or a square root.
Polynomial in numerator and denominator → factor and cancel.
Square root present → multiply by the conjugate.
Neither obvious → build a table of values for confirmation.
Step 4: If you get k/0 where k ≠ 0, the limit does not exist (vertical asymptote).
Tags: limit notation, limit formula, conjugate formula
Limit notation: lim (x→c) f(x) = L
Read as: "the limit of f(x) as x approaches c equals L."
Conjugate multiplication pattern: (√a − b)(√a + b) = a − b²
This eliminates the square root from the expression. After cancellation, substitute the target value into the simplified form.
Substitution result guide:
Real number → that is the limit
0/0 → indeterminate, use another method
k/0 (k ≠ 0) → limit does not exist
Limits are the mathematical engine behind any situation where you need to know a trend without being able to measure the exact endpoint. Engineers use limits to model what happens as friction approaches zero, economists use them to model marginal cost as production scales, and any rate-of-change calculation (speed, growth, decay) is built on limits.
Students often think 0/0 means the limit is zero. It does not. 0/0 is indeterminate, which means you do not have enough information yet and need a different technique.
Students often assume f(c) and lim (x→c) f(x) are the same thing. They can differ. A function can have a hole at c (undefined there) while the limit at c is perfectly well-defined.
Students sometimes only check values from one side. The two-sided limit exists only when both the left-hand and right-hand limits agree. Always check from both directions.
Students sometimes multiply by the conjugate but forget to multiply the denominator as well. You must multiply both top and bottom to keep the expression equivalent.
⚠️ The 0/0 indeterminate form is heavily tested. Expect at least one problem where substitution fails and you must factor or rationalise.
⚠️ Be ready to build a table of values. Even if the algebraic method is faster, some problems ask you to confirm a limit numerically.
⚠️ Know the conjugate technique inside out. Problems involving √x in a limit are a staple of this chapter.
⚠️ The distinction between "the value of f at c" and "the limit of f at c" is a common exam question, often as a conceptual true/false.
True or false: If f(c) is undefined, then lim (x→c) f(x) does not exist.
False. The function can be undefined at c while the limit is well-defined (as in the S(t) example where S(1) = 0/0 but the limit was 4).
Fill in the blank: When substitution gives 0/0, the result is called ______.
Indeterminate.
True or false: To rationalise a fraction with √x in the numerator, you multiply only the numerator by the conjugate.
False. You must multiply both numerator and denominator.
Fill in the blank: lim (x→0) (x² + 5) = ______.
True or false: If the left-hand limit is 3 and the right-hand limit is 4, the two-sided limit is 3.5.
False. When the one-sided limits disagree, the two-sided limit does not exist.
Q: Find lim (x→2) (x² − 4) / (x − 2).
A: Substitution gives 0/0. Factor the numerator: (x − 2)(x + 2) / (x − 2). Cancel (x − 2). You are left with x + 2. Substitute x = 2: the limit is 4.
Q: Find lim (x→0) (x² + 5).
A: Substitution works directly. 0² + 5 = 5. The limit is 5.
Q: Find lim (x→1) (√x − 1) / (x − 1).
A: Substitution gives 0/0. Multiply by the conjugate (√x + 1) / (√x + 1). Simplifies to 1 / (√x + 1). Substitute x = 1: 1 / (1 + 1) = 1/2.
Q: A student plugs t = 1 into S(t) and gets 0/0. They conclude the limit does not exist. What is wrong with this reasoning?
A: 0/0 is indeterminate, not "does not exist." It means you need a different method (table, factoring, conjugate) to find the limit. In this case the limit turned out to be 4.
Q: Explain the difference between f(c) and lim (x→c) f(x).
A: f(c) is the output of the function at exactly x = c. The limit is the value the outputs approach as x gets close to c from both sides. They may be equal, or the function may be undefined at c while the limit exists, or the function value at c may differ from the limit (a removable discontinuity).
This connects directly to derivatives (Ch. 1, Part 2), because the derivative is defined as a limit: f′(x) = lim (h→0) [f(x+h) − f(x)] / h. Every technique you learn here for handling 0/0, especially factoring and conjugate multiplication, will reappear when you compute derivatives from the definition.
Limits also underpin the concept of continuity, which matters later when you use the Intermediate Value Theorem and when you check whether a function is differentiable at a point.
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