Limit Calculator
Find the limit of a function as x approaches a number or infinity, from both sides or just one. See the working (plugging in, L’Hôpital’s rule, highest powers), a table of values closing in from each side and a graph.
Type x^2 for x², sqrt(x) for √x, e^x, ln(x), sin(x) and |x| or abs(x). log(x) is base 10.
A number, a fraction, π, or ∞ (type inf or -inf).
How your limit is worked out
Your function, step by step: try plugging in, deal with any 0/0, then check the values from both sides.
- 1
Try plugging in
sin(0)/0= 0/0undefined: dig deeper
- 2
L’Hôpital’s rule
(sin(x))′ = cos(x)(x)′ = 1→ cos(x)/1
- 3
Plug in again
cos(0)= 1
- 4
Check both sides
−0.001: 0.999999830.001: 0.99999983limit = 1
Read more: the formula, a worked example, L’Hôpital’s rule and one-sided limits
A limit asks what value f(x) heads for as x gets closer and closer to a (or grows towards ∞), whether or not f(a) itself exists. The calculator first tries plugging a in. If that gives 0/0 or ∞/∞ it uses L’Hôpital’s rule; for fractions at infinity it compares the highest powers. Then it checks the answer against a table of values from each side.
The formula
lim (x → a) f(x) = L means f(x) gets as close to L as you like when x is close enough to a (x ≠ a)
continuous at a: lim (x → a) f(x) = f(a), so just plug in
L’Hôpital: if f(a) / g(a) is 0/0 or ∞/∞, then lim f(x)/g(x) = lim f′(x)/g′(x)
two-sided limit exists only if lim (x → a⁻) = lim (x → a⁺)
at ∞: (aₙxⁿ + …) / (bₙxⁿ + …) → aₙ ÷ bₙ
Worked example
Find the limit of sin(x)/x as x → 0. Plugging in gives sin(0)/0 = 0/0, which has no value, so look deeper. It is 0/0, so L’Hôpital’s rule applies: the derivative of the top, sin(x), is cos(x), and the derivative of the bottom, x, is 1. Now plug in: cos(0)/1 = 1/1 = 1. The table agrees: at x = 0.1 the value is 0.99833417, at x = 0.01 it is 0.99998333, and at x = 0.001 it is 0.99999983. The values from the left (x = −0.1, −0.01, −0.001) are exactly the same, because sin(x)/x is an even function. Both sides head for 1, so the limit is 1, even though the function itself is not defined at x = 0 (a removable gap, shown as a hollow circle on the graph).
Indeterminate forms
| Form | What to do | Example |
|---|---|---|
| 0/0 or ∞/∞ | L’Hôpital’s rule (or factor and cancel) | (x² − 4)/(x − 2) → 4 at x = 2 |
| 0 · ∞ | Rewrite as a fraction, then L’Hôpital | x·ln(x) = ln(x)/(1/x) → 0 at 0⁺ |
| 1^∞, 0⁰, ∞⁰ | Take ln, find that limit, then raise e to it | (1 + 1/x)ˣ → e at ∞ |
| ∞ − ∞ | Combine or rationalise (the table shows the value) | √(x² + x) − x → 1/2 at ∞ |
| number/0 | Not indeterminate: it blows up; check the sign on each side | 1/x: −∞ from the left, ∞ from the right |
Continuity
A function is continuous at a when the limit exists and equals f(a). If the limit exists but f(a) is missing or different, there is a removable gap (a hole). If the two sides head for different numbers there is a jump. If a side runs off to ±∞ there is a vertical asymptote. The Continuity box tells you which one you have.
Reading the table
The table puts x at a ± 0.1, 0.01, 0.001 and so on (or 10, 100, 1000… for ∞). When both columns settle on the same number, that is the limit. Very close to a, a computer loses digits when two nearly equal numbers are subtracted, so the last rows can wobble slightly; the steps above use exact algebra wherever they can.
Trig functions use radians. Exact answers come from the algebra, from recognising a simple number such as 1/2, π or e in the decimals, or from the nerdamer CAS when it agrees with the numbers.
Common questions
What is a limit?
The limit of f(x) as x approaches a is the value f(x) gets closer and closer to as x gets closer to a, without x ever being exactly a. sin(x)/x can’t be worked out at x = 0 (it would be 0/0), but at x = 0.001 it is 0.9999998, and it gets ever nearer to 1, so the limit is 1.
When does a limit not exist?
A two-sided limit does not exist when the left and right sides head for different values (1/x goes to −∞ from the left and ∞ from the right), when the values keep wobbling without settling (sin(1/x) near 0), or when the function isn’t defined near the point. Limits that grow without bound are written as ∞ or −∞.
What is L’Hôpital’s rule?
If plugging in gives 0/0 or ∞/∞, differentiate the top and the bottom separately (not as a fraction) and try again. For sin(x)/x at 0: the top becomes cos(x) and the bottom becomes 1, and cos(0)/1 = 1. You can repeat it: (1 − cos(x))/x² needs two rounds and gives 1/2.
What is a one-sided limit?
A left-hand limit (x → a⁻) only uses x values smaller than a, and a right-hand limit (x → a⁺) only larger ones. They matter at jumps and asymptotes. For 1/x at 0, the left-hand limit is −∞ and the right-hand limit is ∞. The two-sided limit exists only when both one-sided limits are equal.
How do you find a limit at infinity?
For a fraction of polynomials, compare the highest powers. If the top and bottom have the same highest power, the limit is the ratio of their leading numbers: (3x² + 1)/(2x² − 5) → 3/2. If the bottom’s power is higher, the limit is 0. If the top’s is higher, it grows without bound.
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