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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?

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) Does Not Exist

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.

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:

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...

?

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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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?

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

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

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

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:

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!!!

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.

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

Pg 59:9­47 Odd