Ways to Evaluate Limits Graphically show graph and

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Ways to Evaluate Limits: • Graphically – show graph and arrows traveling from each

Ways to Evaluate Limits: • Graphically – show graph and arrows traveling from each side of the xvalue to find limit • Numerically – show table values from both the right and left of the xvalue to discover limit • Analytically - algebraically

In AP Calculus, we will be approaching problems in three different ways: q. Analytically

In AP Calculus, we will be approaching problems in three different ways: q. Analytically (using the equation) q. Numerically q. Graphically

Finding Limits Graphically and Numerically

Finding Limits Graphically and Numerically

Finding Limits Graphically The informal definition of a limit is “what is happening to

Finding Limits Graphically The informal definition of a limit is “what is happening to y as x gets close to a certain number”

Notation for a Limit This is read: The limit of f of x as

Notation for a Limit This is read: The limit of f of x as x approaches c equals L.

If we are concerned with the limit of f(x) as we approach some value

If we are concerned with the limit of f(x) as we approach some value c from the left hand side, we write

One-sided Limits—Left-Hand Limit This is read--The limit of f as x approaches c from

One-sided Limits—Left-Hand Limit This is read--The limit of f as x approaches c from the left.

If we are concerned with the limit of f(x) as we approach some value

If we are concerned with the limit of f(x) as we approach some value c from the right hand side, we write

One-sided Limits—Right-Hand Limit This is read: The limit of f as x approaches c

One-sided Limits—Right-Hand Limit This is read: The limit of f as x approaches c from the right.

Definition of a Limit If the righthand lefthand limits are equal and exist, then

Definition of a Limit If the righthand lefthand limits are equal and exist, then the limit exists.

In order for a limit to exist at c = and we write:

In order for a limit to exist at c = and we write:

Example 2 --Graphically Look at the graph and notice that y approaches 2 as

Example 2 --Graphically Look at the graph and notice that y approaches 2 as x approaches 1 from the left. This is also true from the right. Therefore the limit exists and is 2.

Graphically! What causes this discontinuity? ? ? ALGEBRA!

Graphically! What causes this discontinuity? ? ? ALGEBRA!

Simplify:

Simplify:

When we are computing limits the question that we are really asking is what

When we are computing limits the question that we are really asking is what y value is our graph “intending to take” as we move on towards x = 2 on our graph. We are NOT asking what y value the graph takes at the point in question!

Example 2 --Numerically Look at the table and notice that y approaches 2 as

Example 2 --Numerically Look at the table and notice that y approaches 2 as x approaches 1 from the left. (slightly smaller than 1) This is also true from the right. (slightly larger than 1) Therefore the limit Exists and is 2.

Finding Limits EXAMPLE Determine whether the limit exists. If it does, compute it. SOLUTION

Finding Limits EXAMPLE Determine whether the limit exists. If it does, compute it. SOLUTION Let us make a table of values of x approaching 4 and the corresponding values of x 3 – 7 as we approach for both from above (from the right) and below (from the left) x x 3 - 7 4. 1 61. 921 3. 9 52. 319 4. 01 57. 481 3. 99 56. 521 4. 001 57. 048 3. 999 56. 952 4. 0001 57. 005 3. 9999 56. 995 As x approaches 4, it appears that x 3 – 7 approaches 57. In terms of our notation,

One more try…. . • Turn on the TI-83/84 or 89 • Graph y

One more try…. . • Turn on the TI-83/84 or 89 • Graph y = 2 x + 2 • Create a table • Study what happens as x approaches 5. • Make sure your tblset is set to: Independent “ask”. You can then choose any x value you like and get its y-value

From the Left From the Right x x f(x)

From the Left From the Right x x f(x)

Example. Evaluate the following limit: Let’s make a table of values…

Example. Evaluate the following limit: Let’s make a table of values…

From the Left From the Right x x f(x)

From the Left From the Right x x f(x)

Graphically! What causes this discontinuity? ? ? ALGEBRA!

Graphically! What causes this discontinuity? ? ? ALGEBRA!

Simplify:

Simplify:

When we are computing limits the question that we are really asking is what

When we are computing limits the question that we are really asking is what y value is our graph “intending to take” as we move on towards x = 2 on our graph. We are NOT asking what y value the graph takes at the point in question!

Example. Evaluate the following limit:

Example. Evaluate the following limit:

The limit is NOT 5!!! Remember from the discussion after the first example that

The limit is NOT 5!!! Remember from the discussion after the first example that limits do not care what the function is actually doing at the point in question. Limits are only concerned with what is going on around the point. Since the only thing about the function that we actually changed was its behavior at x = 2 this will not change the limit.

Finding Limits EXAMPLE For the following function g (x), determine whether or not exists.

Finding Limits EXAMPLE For the following function g (x), determine whether or not exists. If so, give the limit. SOLUTION We can see that as x gets closer and closer to 3, the values of g(x) get closer and closer to 2. This is true for values of x to both the right and the left of 3.

Limit of the Function • Note: we can approach a limit from • left

Limit of the Function • Note: we can approach a limit from • left … right …both sides • Function may or may not exist at that point • At a • right hand limit, no left • function not defined • At b • left handed limit, no right • function defined a b

Observing a Limit • Can be observed on a graph.

Observing a Limit • Can be observed on a graph.

Observing a Limit • Can be observed on a graph.

Observing a Limit • Can be observed on a graph.

Find each limit, if it exists. In order for a limit to exist, the

Find each limit, if it exists. In order for a limit to exist, the two sides of a graph must match at the given x-value. D. S.

1. 1 Limits: A Numerical and Graphical Approach • Thus for Example 1: •

1. 1 Limits: A Numerical and Graphical Approach • Thus for Example 1: • • does not exist

Non Existent Limits • f(x) grows without bound

Non Existent Limits • f(x) grows without bound

Non Existent Limits

Non Existent Limits

Graphical Example 2 What happens as x approaches zero? The limit as x approaches

Graphical Example 2 What happens as x approaches zero? The limit as x approaches zero does not exist.

Graphical Example 2 What happens as x approaches zero? The limit as x approaches

Graphical Example 2 What happens as x approaches zero? The limit as x approaches zero does not exist.

From this graph we can see that as we move in towards t=0 the

From this graph we can see that as we move in towards t=0 the function starts oscillating wildly and in fact the oscillations increases in speed the closer to t=0 that we get. Recall from our definition of the limit that in order for a limit to exist the function must be settling down in towards a single value as we get closer to the point in question. This function clearly does not settle in towards a single number and so this limit does not exist!

Common Types of Behavior Associated with Nonexistence of a Limit

Common Types of Behavior Associated with Nonexistence of a Limit

SUMMATION Introduction to limits • • The limit of a function is the y

SUMMATION Introduction to limits • • The limit of a function is the y value the graph is getting closer to as x gets closer to a particular value Making a table of values to calculate the limit – must be done on a calculator Sketch a graph to calculate the limit, or use an already existing graph to calculate the limit

When limits fail to exist 1. When the right hand left hand limits do

When limits fail to exist 1. When the right hand left hand limits do not agree 2. When there is unbounded behavior (as we have just seen) 3. When there is oscillating behavior