Removing discontinuities

When studying "Removing discontinuities (factoring)," focus on the following key points:

1. Definition of Discontinuities: Understand what discontinuities are, especially removable discontinuities, which occur when a function is undefined at a certain point but can be made continuous by redefining the function.

2. Factoring Polynomials: Learn how to factor polynomials to identify common factors that may create discontinuities in rational functions.

3. Identifying Removable Discontinuities: Recognize that removable discontinuities can often be found through common factors in the numerator and the denominator of a rational function.

4. Simplifying Expressions: Practice simplifying rational expressions by canceling out common factors, which helps in finding limits, defining function values, and eliminating discontinuities.

5. Finding Limits: Understand how to calculate limits at points of discontinuity by factoring and simplifying the expression.

6. Graphical Interpretation: Be familiar with how factoring and removing discontinuities affect the graph of the function, particularly how it changes the behavior around the point of discontinuity.

7. Application in Calculus: Apply the concept of removing discontinuities when working with limits, derivatives, and integrals in calculus.

By mastering these points, you will effectively understand and apply the concept of removing discontinuities through factoring.

When studying "Removing discontinuities (rationalization)," focus on the following key points:

1. Definition of Discontinuities: Understand what constitutes a discontinuity in a function, specifically in rational functions where division by zero occurs.

2. Identifying Points of Discontinuity: Learn how to find points of discontinuity by examining the denominator of rational functions and solving for when it equals zero.

3. Rationalization Techniques:

   • Factoring: Factor both the numerator and denominator to identify and cancel common factors.
   • Simplification: Simplify the function after canceling factors to remove the discontinuity.
   • Limits: Use limits to analyze the behavior of the function near the discontinuity.

4. Removable vs. Non-removable Discontinuities: Distinguish between removable discontinuities (where the limit exists but the function value does not) and non-removable discontinuities (where the limit does not exist).

5. Graphical Interpretation: Visualize the function before and after rationalization to understand how the graph changes and how discontinuities are affected.

6. Application in Calculus: Recognize the relevance of rationalization when evaluating limits, particularly in cases involving indeterminate forms like $\frac{0}{0}$.

7. Practice Problems: Engage with various exercises that involve identifying discontinuities, rationalizing functions, and applying limits to strengthen understanding.

By focusing on these points, you'll build a solid foundation in understanding and removing discontinuities in rational functions.