Supports: +, -, *, /, ^, sin, cos, tan, ln, log, sqrt, abs, exp, pi, e
Forms that require special techniques:
- Use ^ for exponents: x^2, e^x
- Use / for division: (x-1)/(x+1)
- Functions: sin, cos, tan, ln, log, sqrt
- Constants: pi, e
- Implicit multiplication: 2x = 2*x
In calculus, a limit describes the value that a function approaches as the input (or variable) approaches some value. Limits are fundamental to calculus and mathematical analysis, providing the foundation for defining derivatives and integrals. The concept of a limit allows us to analyze function behavior near points where the function may not be directly defined.
The notation lim (x → a) f(x) = L means that as x gets arbitrarily close to a, the function f(x) gets arbitrarily close to L. This is true whether x approaches a from the left, from the right, or from both sides. Understanding limits is essential for studying continuity, derivatives, and the behavior of functions at boundaries and infinity.
Two-sided Limit
lim (x → a) f(x) exists only when both left and right limits exist and are equal.
One-sided Limits
Left-hand limit (x → a⁻) and right-hand limit (x → a⁺) approach from one direction only.
Limits at Infinity
lim (x → ±∞) f(x) describes the horizontal asymptotic behavior of the function.
Infinite Limits
When f(x) → ±∞ as x → a, describing vertical asymptotic behavior.
Limit calculations follow standard calculus rules. This calculator uses numerical approximation methods and may not handle all algebraic simplifications. Results depend on correct function input and the selected approach value. For complex limits involving indeterminate forms, algebraic manipulation or L'Hôpital's Rule may be required for exact solutions.