Question

Piecewise Function Limit Calculator

A) Limit Analysis Tool
B) Limit Finder for Piecewise Functions
C) Limit Solver for Function Parts
D) Function Boundary Limit Evaluator

Answer 1

To find the limit of a piecewise function, identify the common boundary points, evaluate the limits from both sides of the boundary points, and compare them.

Step-by-step explanation:

Limit of Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each of which is valid over a specific interval. To find the limit of a piecewise function, you need to determine the limit of each sub-function at the common boundary points and compare them.

1. Identify the common boundary points between the sub-functions.
2. Evaluate the limit of each sub-function as the variable approaches the boundary point from both sides.
3. If the limits from both sides are equal, then the overall limit at that point exists and is equal to the common limit value. If the limits from both sides are unequal, then the overall limit at that point does not exist.

Let's consider an example: Given the piecewise function f(x) = { x^2, x < 0; 2x+1, x >= 0 }, we want to find the limit of f(x) as x approaches 0. We evaluate the limits of the sub-functions x^2 and 2x+1 separately:

1. For x < 0, the limit of x^2 as x approaches 0 is 0^2 = 0.
2. For x >= 0, the limit of 2x+1 as x approaches 0 is 2(0)+1 = 1.

Since the limits from both sides are unequal, the overall limit of f(x) as x approaches 0 does not exist.