Learning objectives
Explain how average rates of change over shrinking intervals lead to the idea of a limit, and write limits and one-sided limits in correct notation
Estimate limits from graphs and tables, and evaluate limits of combined functions with the limit laws
Resolve 0/0 forms by factorizing, rationalizing, simplifying complex fractions or using the standard trigonometric limits
Apply the squeeze theorem, and choose and justify a method for any given limit
Key definitions
Limit
lim f(x) = L as x → c means f(x) can be made as close to L as we like by taking x close enough to c, but not equal to c. The value f(c) plays no part.
One-sided limit
The limit as x → c⁻ uses only x-values below c; as x → c⁺ only x-values above c. The two-sided limit exists exactly when both exist and are equal.
Indeterminate form 0/0
A quotient whose numerator and denominator both tend to 0. It gives no answer on its own: the expression must be rewritten before the limit can be found.
Squeeze theorem
If g(x) ≤ f(x) ≤ h(x) for x near c and g and h both tend to L as x → c, then f(x) also tends to L.
Explanation
Why limits? The average rate of change of f over [a, a + h] is (f(a + h) − f(a))/h. As h shrinks, these values settle toward one number, and that number is a limit. We write lim f(x) = L as x → c when f(x) gets arbitrarily close to L for all x close enough to c, from both sides. The value at c does not matter: f(c) may equal L, differ from L or not exist. A one-sided limit uses only x < c (written x → c⁻) or only x > c (x → c⁺). A two-sided limit does not exist when the one-sided limits differ (a jump), when f(x) grows without bound (writing = ∞ describes the behavior, but there is still no limit), or when f oscillates, as sin(1/x) does near 0.
Graphs, tables and the limit laws. On a graph, follow the curve toward x = c from each side and read the y-value it approaches, ignoring any open or filled dot at c itself. A table such as x = 1.9, 1.99, 1.999 only suggests a limit, because it samples finitely many points. If lim f = A and lim g = B, then f ± g → A ± B, kf → kA, fg → AB, fⁿ → Aⁿ and f/g → A/B provided B ≠ 0. For a composition, lim f(g(x)) = f(B) when f is continuous at B; if f is not continuous there, check from which side g(x) approaches B. With graphical data, read each limit separately and then combine them with the laws.
The form 0/0. Always try direct substitution first. A non-zero number over 0 signals unbounded behavior, so examine the sign of each one-sided limit. The form 0/0 means rewrite: factorize and cancel, e.g. (x² − 9)/(x − 3) = x + 3 → 6 as x → 3; multiply by a conjugate, e.g. (√(x + 4) − 2)/x = 1/(√(x + 4) + 2) → 1/4 as x → 0; or combine a complex fraction, e.g. (1/x − 1/3)/(x − 3) = −1/(3x) → −1/9 as x → 3. For trigonometry use lim (sin x)/x = 1 and lim (1 − cos x)/x = 0 as x → 0 (x in radians), rescaling as needed: sin 5x/x = 5 · sin 5x/(5x) → 5.
Squeeze and choosing a method. When algebra fails, trap the function. Since −1 ≤ sin(1/x) ≤ 1, we have −x² ≤ x² sin(1/x) ≤ x² for x ≠ 0, and both bounds tend to 0, so x² sin(1/x) → 0 by the squeeze theorem. On the exam, name the procedure and the reason: "the one-sided limits are both 4, so the limit is 4"; "the form is 0/0, so factorize"; "the bounds have the same limit 0, so by the squeeze theorem…". A good answer moves easily between the graph, a table, the algebra and a sentence describing the same limit.
Formulae
lim x→c f(x) = L ⇔ lim x→c⁻ f(x) = lim x→c⁺ f(x) = L
lim (f ± g) = lim f ± lim g; lim fg = (lim f)(lim g); lim f/g = lim f ÷ lim g if lim g ≠ 0
lim x→0 (sin x)/x = 1; lim x→0 (1 − cos x)/x = 0
Squeeze: g ≤ f ≤ h near c and lim g = lim h = L ⇒ lim f = L
Worked examples
q: Evaluate lim (√(x + 5) − 3)/(x − 4) as x → 4, showing your method.
steps: Direct substitution gives (3 − 3)/0 = 0/0, which is indeterminate, so rewrite the expression.
Multiply top and bottom by the conjugate √(x + 5) + 3: the numerator becomes (x + 5) − 9 = x − 4.
Cancel the common factor (x − 4), valid because x ≠ 4 in a limit: the expression becomes 1/(√(x + 5) + 3).
Substitute x = 4: 1/(3 + 3) = 1/6.
In context
A speedometer reading is a limit: it reports the average speed over an ever-shorter time interval around the current instant.
Engineers use the squeeze idea to bound an error: if a measurement is trapped between two estimates that converge, the true value is pinned down.
Common misconceptions
Thinking the limit as x → c must equal f(c): the limit ignores the value at c entirely.
Treating 0/0 as meaning the limit is 0, 1 or does not exist: it means more work is needed.
Believing a table of values proves a limit: it only suggests one.
Writing a limit of ∞ as though it exists: an infinite limit describes unbounded behavior, and the limit does not exist.
Study tips
Show the substitution that produced 0/0 before you factorize; it justifies the algebra that follows.
When a graph has a jump, write both one-sided limits explicitly and then state whether they are equal.
Check a trigonometric limit is in radians and that the angle matches the denominator, rescaling if needed.
Summary
A limit describes what f(x) approaches near c, not the value at c.
The two-sided limit exists only if both one-sided limits exist and are equal.
0/0 means rewrite: factorize, conjugate, combine fractions or use sin x/x → 1.
Squeeze a function between two bounds with a common limit when algebra cannot help.
Check your understanding
Question 1: lim (x² − 4)/(x − 2) as x → 2 equals…
1. 4
2. 0
3. It does not exist
Question 2: If f(x) → 3 as x → 1⁻ and f(x) → 5 as x → 1⁺, then lim f(x) as x → 1…
1. does not exist
2. equals f(1)
3. equals 4