lim Limit Calculator: Complete Calculus Guide
Master limits with our advanced calculator featuring multiple solution methods including L'Hôpital's rule, factoring, and graphical analysis.
Limit Evaluation Methods
Direct Substitution
Simply plug in the value. Works when function is continuous at that point.
Factoring
Factor numerator/denominator, cancel common terms, then substitute.
Rationalization
Multiply by conjugate to eliminate radicals causing indeterminate forms.
L'Hôpital's Rule
For 0/0 or ∞/∞ forms: take derivative of numerator and denominator.
Common Limits Reference
| Limit | Value | Limit | Value |
|---|---|---|---|
| lim(x→0) sin(x)/x | 1 | lim(x→∞) (1+1/x)^x | e |
| lim(x→0) (1-cos(x))/x | 0 | lim(x→0) (e^x-1)/x | 1 |
| lim(x→0) ln(1+x)/x | 1 | lim(x→∞) e^x | ∞ |
Frequently Asked Questions
Q: What is a limit?
A: A limit describes the value a function approaches as the input approaches some value. Formally: lim(x→c) f(x) = L means f(x) gets arbitrarily close to L as x gets closer to c. Limits are fundamental to calculus, defining derivatives and integrals.
Q: When do you use L'Hôpital's rule?
A: Use L'Hôpital's rule when direct substitution gives 0/0 or ∞/∞ (indeterminate forms). The rule states: lim(x→c) f(x)/g(x) = lim(x→c) f'(x)/g'(x). You may need to apply it multiple times until getting a determinate form.
Q: What's the difference between left and right limits?
A: Left-hand limit (x→c⁻) approaches from values less than c. Right-hand limit (x→c⁺) approaches from values greater than c. For a two-sided limit to exist, both one-sided limits must exist and be equal.
Q: Can a limit exist at a discontinuity?
A: Yes! A limit can exist even if the function is undefined or has a different value at that point. Example: lim(x→1) (x²-1)/(x-1) = 2, even though the function is undefined at x=1. The limit describes behavior near the point, not at the point.