Math 1325 - Lecture 9.1 - Intro to Calculus & Limits

Professor Michael Bailey introduces the course, Math 1325 Calculus for Business and Social Science, at Brookhaven College. He explains two fundamental concepts of calculus: instantaneous change, which is about how things change at a precise moment, and accumulated change, which is about the result of change over time. These concepts are represented by derivatives and integrals, respectively. The lecture also touches on the rate of change from algebra, which is related to the slope of a line. The professor uses the example of the growth of Internet users to illustrate the concept of average rate of change. The idea of a limit is introduced as a method to understand the value of a function as it approaches a certain point, which is crucial for calculating instantaneous rates of change.

The lecture continues with a deeper exploration of limits, emphasizing the difference between the limit value as X approaches a specific point and the function's value at that point. One-sided limits are introduced, showing how the direction from which X approaches the point can affect the limit's value. The concept of a function being continuous is discussed, with examples of how to calculate limits when the function is continuous at a point. Properties of limits are listed, and the importance of understanding these properties for calculating limits is highlighted. The lecture also clarifies the notation and calculation methods for limits, including the limit of a constant, a variable, a sum or difference, a product, a quotient, and the nth root of a function.

The third paragraph delves into the calculation of limits for continuous functions, explaining that if a function is continuous at a point, the limit as X approaches that point is the same as the function's value at that point. The professor provides methods for calculating limits for polynomial, rational, and piecewise functions. Special attention is given to rational functions, where the domain must be considered to ensure the denominator is not zero. The lecture also addresses how to handle cases where the function is discontinuous, such as when there are holes or jumps in the graph. Techniques for factoring and reducing fractions are discussed to find limits in certain scenarios, and the conditions under which a limit does not exist are explained.

The final paragraph focuses on calculating limits for piecewise functions, which have different equations based on the value of X. The process involves checking one-sided limits as X approaches a specific point within the function's domain. The full limit exists only if both one-sided limits exist and are equal. The professor illustrates this with an example, showing how to calculate the limit as X approaches 1 from both the positive and negative directions using the appropriate piecewise function equation for each approach. The lecture concludes with the finding that if the one-sided limits are not equal, the full limit does not exist.