Math 1241, Spring 2014 Section 3.1, Part One Introduction to Limits Finding Limits From a Graph One-sided and Infinite Limits

Conceptual idea of a Limit If I live close enough to campus, I can drive there in a very short amount of time. Intuitively, this is a true statement. However, it’s somewhat ambiguous. Why? What do we mean by… – “close enough” to campus? – a “very short” amount of time?

Conceptual idea of a Limit “Close enough to campus” should mean “within a certain distance.” What distance? “A short amount of time” should mean “less than a certain amount of time.” How much? We must also put these together so that the following statement is true: If I live within ____ miles of campus, I can drive there in less than ____ minutes.

Conceptual idea of a Limit In theory, my commute time is determined by (is a function of) the distance I live from campus. Reality is more complicated, but… If I specify my maximum commute time, could you determine my maximum distance? Could you do this regardless of what maximum time I specify? These questions are related to the precise, mathematical definition of a limit.

Distance versus Travel Time

A more complicated limit Google Maps: It takes 18 minutes to drive 15.5 miles from Turner Field to Clayton State (obviously, this ignores downtown traffic). So…. If I start “close enough” to Turner Field, my driving time to Clayton state is “nearly” 18 minutes? To make this precise, what would you need to specify? (Answers on the next slide)

A more complicated limit You would need to tell me: – Within what distance of Turner Field? – How close to 18 minutes? If I start within ____ miles of Turner Field, my drive time to CSU is within ____ minutes of the 18 minutes claimed by Google Maps. Question: If my drive time is nearly 18 minutes, did I start close to Turner Field?

Distance versus Travel Time

An easy algebraic limit In general, we’ll have a function y = f(x), and ask what happens to the output (y) as the input (x) gets “close to” some fixed value (a). Example: What happens to the value of the function y = 2x - 3 as x gets close to 2? Try to answer this without plugging in x = 2. The reason for this restriction will become clear in later examples. This is a straight line, try drawing a graph!

Example: f(x) = 2x - 3

Another Example

Limit Notation

Graphical Exercise

Some notes about limits The limit of a function must be a single number. This means a particular limit might not exist – Previous example: No limit as x approaches -4. You can often (BUT NOT ALWAYS) evaluate the limit of a “simple” function by plugging in the value x = a. We will discuss when this is permissible in Section 3.2 (Continuity). Although we’ll avoid the formal definition of a limit, but we will introduce algebraic rules for evaluating limits (next time).

One-sided limits

Infinite Limits

Examples of Infinite Limits