What a Limit Means
A limit asks what value f(x) approaches as x gets close to a point, without caring what happens at the point itself. That is why (x² − 1)/(x − 1) has a limit of 2 at x = 1 even though the function is undefined there: it behaves exactly like x + 1 everywhere else.
Cauchy and Weierstrass made this rigorous in the 1800s with the epsilon-delta definition. For practical work you need three habits: try substitution first, simplify if you get 0/0, and check both sides.
The Method Ladder
| Form after substitution | Technique | Example |
|---|---|---|
| A number | Done | lim x→3 of x² + 1 = 10 |
| 0/0 | Factor and cancel, or rationalise | lim x→1 of (x²−1)/(x−1) = 2 |
| 0/0 or ∞/∞ | L’Hôpital’s rule | lim x→0 of (eˣ−1−x)/x² = 1/2 |
| Number/0 | Check each side for ±∞ | lim x→0 of 1/x does not exist |
| x → ∞ with polynomials | Compare leading degrees | lim (3x²+1)/(5x²−2) = 3/5 |
lim x→1 of (x² − 1) ÷ (x − 1)
- Substitution gives 0/0, which is undetermined.
- Factor the top: (x − 1)(x + 1).
- Cancel (x − 1), leaving x + 1.
- Substitute x = 1.
Answer: 2
lim x→0 of (eˣ − 1 − x) ÷ x²
- Substitution gives 0/0. Differentiate top and bottom: (eˣ − 1) ÷ 2x.
- Still 0/0. Differentiate again: eˣ ÷ 2.
- Substitute x = 0.
Answer: 1/2
Famous Limits to Know
| Limit | Value | Why it matters |
|---|---|---|
| sin x ÷ x as x→0 | 1 | Basis of every trig derivative |
| (1 − cos x) ÷ x² as x→0 | 1/2 | Small-angle approximations |
| (1 + 1/n)ⁿ as n→∞ | e ≈ 2.71828 | Definition of e, compound interest |
| (eˣ − 1) ÷ x as x→0 | 1 | Derivative of eˣ at 0 |
| x·ln x as x→0⁺ | 0 | Growth rates: x beats ln x |
Compound interest is the classic case. Invest $1 at 100% interest compounded n times a year and you end with (1 + 1/n)ⁿ: $2.00 at n = 1, $2.61 at n = 12, $2.7145 at n = 365 and never more than e ≈ $2.71828.
One-Sided Limits and Continuity
A two-sided limit exists only when the left and right limits are equal. For |x|/x at 0 the left limit is −1 and the right limit is +1, so the limit does not exist. That is a jump.
A function is continuous at a when three things hold: f(a) is defined, the limit exists, and they are equal. The Continuity mode checks each, then names the failure: removable hole, jump or infinite asymptote.
What We Think Is Overrated, Underrated and Wrong
Limits People Search For
| Search | Answer | Method |
|---|---|---|
| lim x→0 sin x ÷ x | 1 | squeeze theorem |
| lim x→0 (1 − cos x) ÷ x² | 1/2 | half-angle or L’Hôpital twice |
| lim x→0 (eˣ − 1) ÷ x | 1 | derivative of eˣ at 0 |
| lim x→∞ (3x² + 1) ÷ (2x² − 5) | 3/2 | divide by x² |
| lim x→0⁺ x ln x | 0 | rewrite as ln x ÷ (1/x) |
| lim x→0 1/x | does not exist | left −∞, right +∞ |
Original Data: Watching e Appear
| n | (1 + 1/n)ⁿ |
|---|---|
| 10 | 2.59374 |
| 100 | 2.70481 |
| 1,000 | 2.71692 |
| 1,000,000 | 2.71828 |
The limit is e ≈ 2.71828. The values creep up slowly: a hundredfold increase in n adds roughly one more correct digit.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: L’Hôpital’s rule as the first move. Try substitution, factoring or conjugates first; the rule only works on 0/0 or ∞/∞.
Underrated: one-sided limits. They explain every “does not exist” answer.
Plain wrong: saying a limit is “undefined”. 0/0 is an indeterminate form, not an answer, and the limit may be a perfectly good number.
Limit Mistakes That Cost Marks
- Applying L’Hôpital when the form is not 0/0 or ∞/∞.
- Differentiating the quotient instead of top and bottom separately.
- Cancelling terms that are added, not multiplied.
- Ignoring one-sided behaviour at an asymptote.
- Writing ∞ − ∞ or 1^∞ as a value. They are indeterminate.
Why Students Get Stuck on Limits, and What a Good Calculator Should Do
Searching for help with limits usually starts late at night with one problem that will not work out. Plugging in gives 0/0, and the next move is not clear. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to choose between factoring, conjugates and L’Hôpital’s rule, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Stopping at 0/0 and calling it undefinedLimit of (x^2 - 4)/(x - 2) as x approaches 2↗
- Misusing L’Hôpital’s rule when it does not applyLimit of sin(x)/x as x approaches 0↗
- Missing a one-sided limitLimit of (3x^2 + 1)/(x^2 - 5) as x approaches infinity↗
- Not knowing what infinity means in a limitLimit of (1 - cos(x))/x^2 as x approaches 0↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants direct substitution tried first
- Wants the method named when it is not direct
- Wants one-sided limits shown
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Limit of (x^2 - 4)/(x - 2) as x approaches 2↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.