Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

Introduction to Limits

Sometimes you can’t work something out directly … but you can see what it should be as you get closer and closer!

Let’s use this function as an example:

2 1

( )1

xf x

x

And let’s work it out for x=1:

21 1 1 1

1 1 1

0

01

Now 0

0 is a difficulty! We don’y really know the value of

0

0, so we need another way of

answering this. So instead of trying to work it out for x=1 let’s try approaching it closer and closer:

x (x2-1)/(x-1)

0.5 1.50000

0.9 1.90000

0.99 1.99000

0.999 1.99900

0.9999 1.99990

0.99999 1.99999

... ...

Now we can see that as x gets close to 1, then (x2-1)/(x-1) gets close to 2.

We are now faced with an interesting situation:

When x=1 we don't know the answer (it is indeterminate) But we can see that it is going to be 2

We want to give the answer "2" but can't, so instead mathematicians say exactly what is going on by using the special word "limit"

The limit of (x2-1)/(x-1) as x approaches 1 is 2

And it is written in symbols as:

So it is a special way of saying, "ignoring what happens when you get there, but as you get closer and closer the answer gets closer and closer to 2"

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

As a graph it looks like this:

So, in truth, you cannot say what the value at x=1 is.

But you can say that as you approach 1, the limit is 2.

Test Both Sides!

It is like running up a hill and then finding the path is magically "not there"...

... but if you only check one side, who knows what happens?

So you need to test it from both directions to be sure where it "should be"!

So, let's try from the other side: x (x2-1)/(x-1)

1.5 2.50000

1.1 2.10000

1.01 2.01000

1.001 2.00100

1.0001 2.00010

1.00001 2.00001

... ...

Also heading for 2, so that's OK

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

1. SAMPLE PROBLEMS

When it is different from different sides

What if we have a function "f(x)" with a "break" in it like this:

This is a function where the limit does not exist at "a" ... !

You can't say what it is, because there are two competing answers:

3.8 from the left, and 1.3 from the right

But you can use the special "-" or "+" signs (as shown) to define one sided limits:

the left-hand limit (-) is 3.8 the right-hand limit (+) is 1.3

And the ordinary limit "does not exist"

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES 2.

Let:

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

Right-handed limit We say

provided we can make f(x) as close to L as we want for all x sufficiently close to a and x>a without actually letting x be a.

Left-handed limit

We say

provided we can make f(x) as close to L as we want for all x sufficiently close to a and x<a without actually letting x be a.

Fact Given a function f(x) if,

then the normal limit will exist and

Likewise, if

then,

This fact can be turned around to also say that if the two one-sided limits have different values, i.e.,

then the normal limit will not exist.

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES 3. Estimate the value of the following limits.

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES From the graph of this function shown below,

we can see that both of the one-sided limits does not settle down to a single number on either side of . Therefore, neither the left-handed nor the right-handed limit will exist in this case. PROPERTIES OF LIMITS

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

SAMPLE PROBLEMS

1. Evaluate the following limit: 2

22

4 12lim

2x

x x

x x

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

2. Evaluate the following limit: 9

3lim 12

9x

x

x

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES THE SQUEEZE (SANDWICH) THEOREM

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

SAMPLE PROBLEM

Evaluate the limit: 0

sin(2 )limx

x

x

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES Continuity at a Point and on an Open Interval

In Calculus, the term continuous has much the same meaning as it has in everyday usage (no interruption, unbroken, no holes, no jumps, no gaps).

Let’s take a look at the following graphs and “discuss” possible continuity or discontinuity at x=0:

a) HOLE b) JUMP c) HOLE-II c) CONTINUOUS

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES SAMPLE PROBLEM

1. Let’s discuss the continuity of the given function f(x).

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES 2. Let’s discuss the continuity

for the Greatest Integer function:

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES 3. Let’s discuss the continuity of y=f(x) on the interval [-1, 3]:

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES 4. The graph to the right shows a

representation of the functions g(x), f(x), and h(x).

2( )g x x

2 1( ) sinf x x

x

2( )h x x Let’s discuss the continuity of m(x) given by

2 1sin 0

( )

0 0

x xm x x

x

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES The Intermediate Value Theorem

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES 5. Is any real number exactly 1 less than its cube? (Application of the IVT)

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES CALCULATOR ACTIVITY

For each of the following rational functions, determine the limit as x approaches 1 from the left and from the right. (YOU MAY WANT TO SKETCH EACH FUNCTION IN YOUR GC, BUT YOU ARE EXPECTED TO KNOW HOW TO DO THIS WITHOUT USING ONE)

1. 1

( )1

f xx

2. 2

1( )

1f x

x

3.

2

1( )

1f x

x

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES SAMPLE PROBLEMS

(Take notes in your notebook)

1. sin( )

limx

x

x

2. sin( )

limx

x

x

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

3. 2

25

4lim

25x

x

x

4.

2

25

4lim

25x

x

x

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES

5. 2

2

4lim

25x

x

x

6.

2

2

4lim

25x

x

x

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES 7.

Mrs. Cisnero, AP CALCULUS BC CHAPTER 1 NOTES 8.