Math 1241, Spring 2014Section 3.1, Part One

Introduction to LimitsFinding Limits From a GraphOne-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

• What happens to the value of the function

as x gets close to the value of 2?• In this case, we cannot simply plug in x = 2.• However, if x is not equal to 2, the above

expression can be simplified algebraically.• Alternatively, we can draw a graph.

𝑓 (𝑥 )=𝑥2−4𝑥−2

Limit Notation

• We use the following set of symbols:

• Read this as, “The limit, as x approaches a, of f(x) is equal to L.”

• Informally, this means: If the value of x is “close enough” BUT NOT EQUAL to a, then the value of y is “close” (possibly equal) to the number L.

• In the previous examples, we had:

Graphical Exercise

For the function f(x) shown to the right, find…• = _____• = _____• = _____• = _____Pay attention to the open and closed dots!

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

• Question: What is the value of the following?

• Note that we cannot plug in x = 0. It may be helpful to draw a graph.

• Fill in the blank: The value of f(x) is close to _____ whenever x is close to 0.

= ??

One-sided limits

• We can use the following notation:

• The first is a left-sided limit. As x approaches 0 from the left, the function value is close to -1.

• The second is a right-sided limit. As x approaches 0 from the right, the function value is close to 1.

• When the left and right limits are not equal, the ordinary, two-sided limit DOES NOT EXIST.

Infinite Limits

• If x is close to zero, then the function is close to what number? Here is the graph:

Infinite Limits

• IMPORTANT: DOES NOT EXIST!!!• There is a reason for this. As x approaches 0,

the function value keeps getting larger, and never approaches any particular value.

• Notation: • But you CANNOT treat the infinity symbol as

though it were an ordinary number.

Examples of Infinite Limits

Convince yourself (possibly by drawing a graph) that the following are true:

For the left-sided limit, the means that the function value continues to decrease, and does not approach any particular value.

𝑓 (𝑥 )=1𝑥