Limits Calculator
Evaluate limits of functions as x approaches a finite value or infinity. Apply direct substitution for polynomials, find horizontal asymptotes, and identify vertical asymptotes.
∫ Calculus📐 lim(x→a) f(x)🔢 Math
Limit type
Coefficient c
Exponent n
Approach point a
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Limits Calculator | — | lim(x→a) f(x) | numeric or ±∞ |
Step-by-Step Examples
Example 1
Direct sub
lim(x→2)[3x²]
- Polynomial — substitute directly
- 3(4)=12
✓ = 12
Example 2
Infinity
lim(x→∞)[1/x²]
- 1/x²→0; horizontal asymptote y=0
✓ = 0
Example 3
Vertical asymptote
lim(x→0)[1/x]
- →+∞ from right, →-∞ from left
- No two-sided limit: vertical asymptote
✓ Does not exist
Real-World Applications
Calculus Foundation
Derivatives and integrals are both defined using limits.
Asymptotes
Rational function behavior via limits.
Physics
Instantaneous velocity = lim(Δt→0)[Δx/Δt].
Finance
Compound interest: lim(n→∞)(1+r/n)^n=eʳ.
Common Mistakes to Avoid
⚠️
Limit ≠ function value
lim can exist even if f(a) undefined (holes in graphs).
⚠️
One-sided limits
If left ≠ right limit, two-sided limit does not exist.
⚠️
0/0 is indeterminate
Factor, cancel, or use L'Hôpital's rule.
Frequently Asked Questions
What is a limit? ▾
Value f(x) approaches as x→a. lim(x→a)f(x)=L.
When to substitute directly? ▾
For continuous functions. If 0/0 results, use other methods.
L'Hôpital's rule? ▾
For 0/0 or ∞/∞: lim f/g = lim f'/g'.
Limit definition of derivative? ▾
f'(x)=lim(h→0)[f(x+h)-f(x)]/h.
Common limits? ▾
lim sin(x)/x=1 as x→0. lim(1+1/x)^x=e as x→∞.
Related Math Calculators
Formula Explorer connections
Interpretation: This formula measures limiting behavior, instantaneous change, accumulation or multivariable variation. Assumption: The function must satisfy the continuity or differentiability conditions required by the method. Check domains, bounds, orientation and singularities.