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1.5 Limits (Welcome to Calculus)

Given:

Translate: "What is f(6)?"

What is the value of the function f when x = 6?

Translate: "What is ?"

What value is the function approaching as x gets very close to 6?

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Algebra studies what is.

Calculus studies what is happening!!

If I want the value of y for a given x, I evaluate the function, ie f(6). This gives me a point on the curve.

The limit describes the value that y approaches as we approach a certain x value. (And we approach from very, very close with teeny­tiny steps!!)

A limit exists at x = c when a function's value (y) approaches the same number as x gets closer to c from both the left and right.

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Perhaps a visual will assist you in this concept.

From a previous lesson.

I will graph the original in my calculator:

It looks like an unbroken line, but WE know there is a hole. The table will show this.

So we know that f(­1) is UNDEFINED!!

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But what value of y is the line approaching as we get closer to x = ­1 from the left and right?

(What is ?)

The table will help again, but now the increments of x will be smaller (much smaller)

What value is y approaching from the left?

From the right?

So I'll ask again, what is ?

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In order for a limit to exist at a particular x­value, the graph must converge (approach) to the same y value from both sides.

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In previous example, the function was undefined at x = ­1, but the limit still existed.

Any combination is possible:

Function defined, Limit existsFunction undefined, Limit existsFunction defined, Limit DNEFunction undefined, Limit DNE

Times when one might have an answer and the other does not:

A) Holes

B) Boundaries of Piecewise Functions

(More?)

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Defined

Exists

Abe approaches from the left... Ulysses approaches from the right...

Function Defined; Limit Exists

The club is there, and they meet.

A real­life example of the different combinations, using two friends and a club:

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Exists

Function Undefined; Limit Exists

Abe approaches from the left... Ulysses approaches from the right...

The club exploded, but they approach the leftover hole.

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DNE

Defined

Function Defined; Limit DNE

The club is there, and Abe went to the club, but Ulysses got lost.

Abe approaches from the left... Ulysses approaches from the right...

?

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September 24, 2015

DNE

Function Undefined; Limit DNE

The club exploded, but Abe showed up. Ulysses got lost.

Abe approaches from the left... Ulysses approaches from the right...

?

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September 24, 2015

Guidelines for evaluating a limit of a NON­piecewise function.

Plug in the value of x we are concerned with. We will have a few possible outcomes.

1. We get a real number back. This is our answer.

2. We get a non­zero divided by zero. This represents an asymptote, and the limit DNE.

3. We get 0/0. This often represents a hole. The limit might exist!

a) Factor and cancel?

b) Multiply by conjugates?

4. We get ∞/∞. This is undefined. The limit might exist!

a) Factor and cancel?

b) Multiply by conjugates?

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Evaluate the following limits:

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x

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Guidelines for evaluating a limit of a piecewise function.

If c (the value of x we are concerned with) is not at a boundary, follow the guidelines for non­piecewise.

If c does fall at a boundary...

Make sure the function is defined immediately to the left and right of c. If not, limit DNE.

Plug in values of x immediately left and right of c (so immediate that the boundary itself works).

If the function's values approach the same number from both sides, the limit is that number.

Else: DNE

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In order for to exist, the following must be true...

This says that for the limit to exist at c, the left sided limit at c must equal the right sided limit at c.

This also allows us to evaluate One­Sided Limits

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Evaluate the following limits:

where

where

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Evaluate the following limits:

Notice that the function is not defined at x = ­2, but the limit still exists. It is possible for a function to have a limit at c that is different from f(c).

where

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Evaluate the following quantities:

Here, f(0) = 5, but the limit as x approaches 0 is ­4.

Given:

Find: and

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Evaluate the following limit:

Here, with direct substitution yielding 0/0, we try factoring or multiplying by conjugates. Neither works, so we look at the one­sided limits. Examining the graph and/or table helps.

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Examination of the graph/table shows the following:

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What about algebraically?

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These work for any non­zero constant c, and they make sense.

Operations with infinity and zero that we need to know about!!!

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Operations with infinity and zero that we need to know about!!!

The following are all undefined; they don't always make sense.

(Undefined and DNE are not the same thing. An undefined answer CAN exist, it just won't be the same thing every time!!)

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Horizontal Asymptote (Precalc Review):

y = 0: if the degree of the numerator is less than the degree of the denominator. (degree = largest exponent)

y = a/b: if the degrees are equal, where a is the leading coefficient of the numerator and b is the leadingcoefficient of the denominator.

DNE: if the degree of the numerator is larger than thedegree of the denominator.

Limits at Infinity:

See above (works for pos & neg infinity).This is a shortcut to the answer; you will be required to back it up with calculus.

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Review: Find the Horizontal Asymptote. New: Find the limit.

Limits at Infinity

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We need to JUSTIFY the answers for limits at infinity. The process is shown below. It involves factoring!!

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Properties of Limits

where

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Day 1Pg 57:1­57 E.O.O.

Day 2Pg 57: 3­59