How to Use This Tool
- Select a Function: Choose from different types of functions to explore various limit behaviors
- Adjust Limit Point: Use the slider to change where you're evaluating the limit
- Observe the Graph: Watch how the function behaves near the limit point
- Read the Analysis: Check left-hand limit, right-hand limit, and function value
- Check Continuity: See if the function is continuous or discontinuous at that point
What is a Limit?
A limit describes the value that a function approaches as the input (x) approaches some value. It's fundamental to calculus and helps us understand function behavior, even at points where the function might not be defined.
One-Sided Limits
Sometimes we need to consider what happens when approaching from different directions:
- Left-hand limit (x→a⁻): The value f(x) approaches as x approaches a from the left (smaller values)
- Right-hand limit (x→a⁺): The value f(x) approaches as x approaches a from the right (larger values)
- Limit exists: Only when both one-sided limits exist and are equal
What is Continuity?
A function is continuous at a point if you can draw it without lifting your pen. Mathematically, a function f(x) is continuous at x = a if:
- f(a) is defined: The function has a value at x = a
- limx→a f(x) exists: The limit as x approaches a exists
- limx→a f(x) = f(a): The limit equals the function value
All three conditions must be satisfied for continuity!
Types of Discontinuities
- Removable Discontinuity (Hole): The limit exists but doesn't equal the function value (or the function is undefined). Can be "fixed" by redefining the function at that point. Example: f(x) = (x²-4)/(x-2) at x=2
- Jump Discontinuity: Left and right limits exist but are different. The function "jumps" from one value to another. Example: step functions, piecewise functions
- Infinite Discontinuity (Vertical Asymptote): The function approaches infinity as x approaches the point. Example: f(x) = 1/x at x=0
Why Limits Matter
Limits are the foundation of calculus:
- Derivatives: The derivative is defined as a limit of the difference quotient
- Integrals: Definite integrals are limits of Riemann sums
- Real-world applications: Instantaneous velocity, instantaneous rate of change, optimization
- Understanding behavior: Limits help us analyze what happens at boundaries and singularities
Limit Properties
If limx→a f(x) = L and limx→a g(x) = M, then:
