Determine the following limits.


a. lim z→3^+ (z − 1)(z − 2) / (z − 3)

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit as z approaches 3 from the right (3+), which indicates we are looking at values of z that are slightly greater than 3.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only. The notation z→3^+ indicates that we are considering the limit as z approaches 3 from the right. This is crucial for understanding the behavior of functions that may have different values or undefined points at the limit point.

Factoring and Simplifying Expressions

Factoring and simplifying expressions is a key technique in calculus for evaluating limits, especially when direct substitution leads to indeterminate forms like 0/0. In the given limit, the expression (z − 1)(z − 2) / (z − 3) can be analyzed by substituting values close to 3 to determine the limit's value, or by simplifying the expression if possible to avoid division by zero.