🎓 Mathematics • Calculus
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Limit Calculator
Evaluates limits of rational expressions as x approaches c, detecting 0/0 indeterminate forms using L’Hôpital’s rule.
Rational Function Limit Calculator: lim (x → c) [N(x) / D(x)]
Calculates limits using direct substitution, L'Hôpital's rule for 0/0, and one-sided limits.
Presets:
Numerator: N(x) = ax² + bx + c
x² +x +
Denominator: D(x) = dx + e
x +
As x Approaches (c):
Limit Value: lim (x → 2)
0
Left-Hand Limit (x → 2⁻)0
Right-Hand Limit (x → 2⁺)0
Evaluation Method & Proof
Indeterminate form 0/0 detected. Applying L'Hôpital's Rule: lim N'(x)/D'(x) = (2(2) + -4) / 1 = 0
What is a Limit in Calculus?
A limit examines what value a function approaches as its input gets arbitrarily close to a specified point.
Limits define both the derivative and the definite integral, forming the rigorous foundation of calculus.
Formula & Step-by-Step Calculation
lim (x → c) [f(x) / g(x)] = lim [f'(x) / g'(x)] (if 0/0)
L’Hôpital’s rule resolves 0/0 or ∞/∞ indeterminate limits by differentiating numerator and denominator.
Worked Step-by-Step Examples
Example 1
lim (x → 2) (x² - 4) / (x - 2)
Solution: 4
• 0/0 indeterminate. Differentiate: 2x / 1. At x = 2, 2(2)/1 = 4.
Common Real-World & Academic Use Cases
- ✓ Evaluating continuity and asymptotic boundaries
- ✓ Deriving instantaneous rates of change
- ✓ Infinite series convergence
How to Use the Limit Calculator
1
Input Expression
Fill in numerator and denominator coefficients.
2
Set Target Point c
Specify what x approaches.
3
View Limit
Inspect two-sided, left-hand, and right-hand limit values.
Frequently Asked Questions
Q: When does a two-sided limit exist?
When both the left-hand limit and right-hand limit exist and are equal.