Evaluate Limit Calculator

Limit Definition:

\[ \text{Limit} = \lim_{x \to a} f(x) \]

Function f(x):

e.g., x^2, sin(x), (x-1)/(x^2-1)

Approaches (a):

value or 'inf' for infinity

Direction:

Limit Result:

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1. What is a Limit?

The limit of a function describes the value that the function approaches as the input approaches some value. Limits are fundamental to calculus and mathematical analysis.

2. How Does the Calculator Work?

The calculator evaluates limits using the mathematical definition:

\[ \text{Limit} = \lim_{x \to a} f(x) \]

Where:

\( f(x) \) — The function to evaluate
\( a \) — The value that x approaches
The limit can be two-sided or one-sided (left/right)

Explanation: The calculator numerically approximates the value that f(x) approaches as x gets arbitrarily close to a.

3. Importance of Limits

Details: Limits are essential for defining derivatives, integrals, and continuity. They're used in physics, engineering, and economics to model instantaneous rates of change and behavior near points.

4. Using the Calculator

Tips: Enter a valid mathematical function (use standard notation), the approaching value (can be a number or infinity), and select the direction if needed.

5. Frequently Asked Questions (FAQ)

Q1: What functions can I input?
A: Polynomials, trigonometric functions (sin, cos, tan), exponential/logarithmic functions, and rational functions.

Q2: How are infinite limits handled?
A: Use 'inf' for infinity. The calculator will evaluate behavior as x grows without bound.

Q3: What if the limit doesn't exist?
A: The calculator will indicate if the left and right limits disagree or if the function oscillates.

Q4: Can I evaluate limits at discontinuities?
A: Yes, the calculator evaluates the limiting behavior, not the function value at the point.

Q5: How precise are the calculations?
A: The calculator uses numerical methods to approximate limits with high precision.