Question

Write a simplified function that coincides with the provided function at all points except one. Utilize a graphing utility to verify your outcome. Consider the limit as x approaches 0 for the function 1/3x-1/3/x.

Answer 1

To create a simplified function from 1/3x - 1/3/x that coincides with it except at one point, you should combine the terms to get (x - 1)/(3x). Use a graphing utility to confirm that they match at all points except x = 0, where the original function is not defined.

Step-by-step explanation:

To simplify the given function f(x) = 1/3x - 1/3/x and find a function that coincides with it at all points except one, we can combine the terms over a common denominator which is 3x. This yields:

f(x) = (x - 1)/(3x)

Graphing utilities such as a graphing calculator can help you verify that this simplified function matches the original function at all points except for x = 0. When you evaluate the limit of f(x) as x approaches 0, you can observe that the function approaches negative infinity, which indicates a vertical asymptote at x = 0.

The point of discrepancy is at x = 0 because division by zero is undefined, resulting in a point of discontinuity for the function. Everywhere else, for x ≠ 0, the original function and the simplified one are equivalent.