Limit Calculator
Evaluate limits, understand one-sided limits, and explore special limit cases with step-by-step explanations.
Use ^ for exponents, * for multiplication, / for division
Enter a number, "infinity", or "-infinity"
Quick Examples:
Understanding Limits
In calculus, a limit is the value that a function approaches as the input approaches some value. Limits are essential for defining continuity, derivatives, and integrals—the fundamental concepts of calculus.
\[ \lim_{x \to a} f(x) = L \]
This notation means "the limit of f(x) as x approaches a equals L." In other words, the value of f(x) gets arbitrarily close to L as x gets arbitrarily close to a.
Types of Limits
- Two-sided limit: The standard limit where x approaches a value from both directions.
- One-sided limit: The limit where x approaches a value from only one direction (left or right).
- Limit at infinity: The limit where x approaches positive or negative infinity.
- Infinite limit: The limit where the function value approaches infinity.
Limit Properties
Sum Rule
\[ \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \]
Product Rule
\[ \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \]
Quotient Rule
\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \] (if $$\lim_{x \to a} g(x) \neq 0$$)
Power Rule
\[ \lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n \]
Techniques for Evaluating Limits
| Technique | When to Use |
|---|---|
| Direct Substitution | When the function is continuous at the point |
| Factoring | For limits with polynomial expressions that can be simplified |
| Rationalization | For limits involving square roots |
| L'Hôpital's Rule | For indeterminate forms like 0/0 or ∞/∞ |
| Squeeze Theorem | When a function is bounded between two functions with the same limit |
Continuity and Limits
A function f(x) is continuous at x = a if and only if:
1. f(a) is defined
2. $$\lim_{x \to a} f(x)$$ exists
3. $$\lim_{x \to a} f(x) = f(a)$$
Limit Calculator Answers
What does the limit calculator do?
The limit calculator evaluates the limit of a function as it approaches a specific point. It helps solve problems involving continuity, derivatives, and asymptotic behavior.
Can it handle one-sided limits?
Yes. You can specify whether you want the left-hand limit, right-hand limit, or two-sided limit, and the calculator will evaluate it accordingly.
What types of functions can it solve?
It works with rational, polynomial, trigonometric, exponential, logarithmic, and piecewise functions—whether the limit exists or not.
Does it show step-by-step solutions?
Yes. Many limit calculators provide a breakdown of steps including factoring, rationalization, or using L'Hôpital’s Rule when applicable.