Identify potential vertical asymptotes by setting the denominator to zero and find common factors for removable discontinuities. Simplify using factorization and cancellation, then evaluate to understand the expression's continuity.
To identify and analyze the discontinuities of rational expressions, examine the denominator for values that make it equal to zero, as these points result in undefined expressions.
These values are potential points of discontinuity, known as vertical asymptotes. Additionally, look for common factors in the numerator and denominator to identify removable or jump discontinuities.
To address or simplify these discontinuities, factorize the rational expression and cancel common factors. For vertical asymptotes, set the denominator equal to zero and solve for x to determine the values causing the discontinuity.
If a factor cancels out in both numerator and denominator, it indicates a removable discontinuity, which can be addressed by simplifying the expression. Evaluate the function at the points of discontinuity to determine the behavior around these points, aiding in a comprehensive analysis of the rational expression's continuity.
Complete question:
How can one identify and analyze the discontinuities of rational expressions, and what strategies can be employed to address or simplify them?