Limits Calculator (with L'Hôpital's Rule)
Enter f(x) and a point a or ∞ to estimate two-sided and one-sided limits, with automatic 0/0 and ∞/∞ indeterminate-form detection.
Input Data
Results
At a glance:The limit lim_{x→a} f(x) = L means f(x) approaches L as x approaches a. If substitution gives 0/0 or ∞/∞, it is an indeterminate form solvable by L'Hôpital's rule (differentiate numerator and denominator) or algebra. For x→∞ observe asymptotic behavior.
Formula
Direct substitution: if f(a) is defined and continuous, limit = f(a).
0/0 or ∞/∞: L'Hôpital lim f/g = lim f'/g' (repeat if needed).
x→∞: compare highest powers, or use logarithms.
$$\lim_{x\to a} f(x) = L$$$$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)} \quad (0/0,\ \infty/\infty)$$How to Use
- Enter f(x) (use x, + - * / ^, sin/cos/tan/exp/ln/sqrt, constants e, pi).
- Enter the approach point a, or 'inf' for infinity.
- The tool gives two-sided and one-sided limits and flags indeterminate forms in the note.
- Compare with the case studies to check your limit working.
Common limits
| Limit | Value | Type |
|---|---|---|
| lim_{x→0} sin x / x | 1 | 0/0 |
| lim_{x→0} (1−cos x)/x² | ½ | 0/0 |
| lim_{x→∞} (1+1/x)^x | e | 1^∞ |
| lim_{x→0} ln(1+x)/x | 1 | 0/0 |
| lim_{x→∞} e^(−x) | 0 | to 0 |
Forms like 1^∞ can be handled by taking logarithms then L'Hôpital.
Case Studies
lim_{x→1} (x²−1)/(x−1)
Substituting x=1 gives 0/0 (indeterminate).
Factor: (x−1)(x+1)/(x−1) = x+1 → limit = 2.
L'Hôpital: derivatives give 2x/1 → at x=1 is 2.
lim_{x→0} sin x / x
0/0 indeterminate.
L'Hôpital: cos x / 1 → at x=0 is 1.
Also from the Taylor series sin x ≈ x, directly 1.
lim_{x→∞} (3x²+1)/(2x²+5)
∞/∞ indeterminate.
Compare leading terms: 3x²/2x² → 3/2 = 1.5.
lim_{x→0} (1−cos x)/x²
0/0 indeterminate.
L'Hôpital twice: → sin x/(2x) → cos x/2 → 1/2 = 0.5.
lim_{x→∞} e^(−x)
Decays to 0 as x grows; at x=10 ≈ 4.5×10⁻⁵.
Limit = 0 (horizontal asymptote y=0).
lim_{x→0} ln(1+x)/x
0/0 indeterminate.
L'Hôpital: 1/(1+x)/1 → at x=0 is 1.
FAQ
What is a 0/0 indeterminate form?
Substitution makes numerator and denominator both 0; the limit is undecided and needs factoring or L'Hôpital. The tool flags it and gives the numeric limit.
When can I use L'Hôpital's rule?
Only for 0/0 or ∞/∞: differentiate numerator and denominator, then take the limit, repeating if still indeterminate.
What if left and right limits differ?
If left ≠ right (step, absolute-value cusp) the two-sided limit does not exist. The tool shows both and warns.
How is x→∞ computed?
The tool samples at x=10², 10⁴, 10⁶… to observe convergence; rational functions compare leading powers, exponentials/logarithms their growth rates.
Why small errors in the result?
Limits are estimated numerically at 10⁻ᵉ scale; the last digits may have float error. Read integer/simple-fraction values exactly.
Which functions are supported?
x, arithmetic, ^, parentheses, and sin/cos/tan/exp/ln/sqrt/abs with constants e, pi.
Can I use abs or piecewise?
abs(x) expresses absolute value; for piecewise functions compute left/right limits separately and compare.
Relation to continuity?
If f is continuous at a, lim_{x→a} f(x) = f(a); a limit exists but differs from (or lacks) the function value at a is a removable discontinuity.
Related Tools
References
Content review: Calculatorism Science Team. Limit numeric estimation and indeterminate-form detection logic verified. Results are for reference only.