Calculate limits of mathematical functions as x approaches a specific value. Enter a function and the approaching value to get the limit with explanations.
What are Limits?
A limit describes the behavior of a function as the input approaches a particular value. Limits are fundamental to calculus and help define derivatives and integrals.
Types of Limits
- Two-sided limit: lim[x→a] f(x) - approaches from both directions
- Left-hand limit: lim[x→a⁻] f(x) - approaches from the left
- Right-hand limit: lim[x→a⁺] f(x) - approaches from the right
- Infinite limit: lim[x→∞] f(x) - as x approaches infinity
Common Limit Rules
- Sum Rule: lim[f(x) + g(x)] = lim f(x) + lim g(x)
- Product Rule: lim[f(x) · g(x)] = lim f(x) · lim g(x)
- Quotient Rule: lim[f(x)/g(x)] = lim f(x) / lim g(x) (if lim g(x) ≠ 0)
- L'Hôpital's Rule: For 0/0 or ∞/∞ forms
Standard Limits
- lim[x→0] sin(x)/x = 1
- lim[x→0] (1-cos(x))/x² = 1/2
- lim[x→∞] (1 + 1/x)^x = e ≈ 2.718
- lim[x→0] (eˣ - 1)/x = 1
Applications
- Defining derivatives and continuity
- Analyzing function behavior
- Solving indeterminate forms
- Understanding asymptotes
