Rational Function Continuity Checker

Understanding Function Continuity

In simple terms, a function is continuous at a point if you can draw its graph near that point without lifting your pen from the paper. For rational functions, which are ratios of polynomials, continuity is generally straightforward except where the denominator becomes zero.

A rational function $$ f(x) = \frac{P(x)}{Q(x)} $$ is continuous at a point 'a' if two main conditions are met when we consider approaching 'a':

• Limit Exists: As 'x' gets closer and closer to 'a' from both sides, the value of $$ f(x) $$ approaches a specific number.
• Limit Equals Function Value: This number that $$ f(x) $$ approaches (the limit) must be the same as the value of the function at $$ x = a $$, i.e., $$ f(a) $$.

For rational functions, potential issues with continuity only occur where the denominator $$ {Q(x)} $$ is zero. At all other points, rational functions are continuous. This tool numerically checks if the limit exists and effectively equals the function value at the given point to determine continuity.

Example

Consider $$ f(x) = \frac{x ^ (2 - 1)}{x - 1} $$. Is it continuous at $$ x = 1 $$? Using the tool, input '(x^2-1)/(x-1)' and point '1'. The result will be 'Not Continuous' because at $$ x = 1 $$, the denominator is zero, leading to a discontinuity, even though the function simplifies to $$ x+1 $$ for $$ x \neq 1 $$.

For deeper understanding, explore resources on calculus and limits on platforms like Khan Academy and Paul's Online Math Notes.