Why is (x²−a²)/(x−a) undefined at x=a?
Denominator is zero, but the limit still exists after canceling.
Limit Calculator
Compute educational limits for difference quotients, polynomials, simple rationals, or decay at infinity.
Choose a limit type
Select the pattern and enter the needed parameters.
About this calculator
How to use Limit Calculator, plus the formulas and logic behind the results.
Tool description
Limits describe the value a function approaches. Removable holes like (x²−a²)/(x−a) simplify to 2x → 2a. At infinity, c/x^n → 0 for n > 0.
How to use
- Select the limit pattern.
- Enter a and any coefficients used by that mode.
- Select Calculate.
- Read the limit value and simplification note.
Formulas & reference
lim x→a (x²−a²)/(x−a) = 2a (x ≠ a)lim x→a polynomial = polynomial(a)lim x→a (px+q)/(rx+s) = (pa+q)/(ra+s) if denominator ≠ 0lim x→∞ c/x^n = 0 for n > 0
Patterns
- (x−a)(x+a)/(x−a) → x+a → 2a
- Polynomials are continuous everywhere
- Watch for zero denominators
- Positive powers of x dominate constants at ∞
Calculation logic
Applies algebraic simplification or direct substitution for each educational case; flags undefined denominators.
Not a general CAS. Only the listed patterns are supported.
Limit Calculator FAQ
Common questions about this calculator and how to use the results.
What if the rational denominator is zero at a?
The two-sided finite limit may fail; the tool reports undefined.
Does infinity mode support n ≤ 0?
n must be positive so the expression decays to 0.
Is this a full limit solver?
No—only selected textbook patterns.
Can a be negative?
Yes.
What about left/right limits?
This tool reports the common two-sided value when it exists for these forms.