Limits FGuilbert ORRHS Two Basic IDEAS of Calculus

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Limits FGuilbert ORRHS

Limits FGuilbert ORRHS

Two Basic IDEAS of Calculus 1. Find the slope of a curve at a

Two Basic IDEAS of Calculus 1. Find the slope of a curve at a point FGuilbert ORRHS

2. Finding the area under a curve. FGuilbert ORRHS

2. Finding the area under a curve. FGuilbert ORRHS

You need to know about limits before you study those ideas. FGuilbert ORRHS

You need to know about limits before you study those ideas. FGuilbert ORRHS

What is a Limit? It is the intended height of a Function. What is

What is a Limit? It is the intended height of a Function. What is the function approac FGuilbert ORRHS

How do you determine a Lim Numerically Graphically Substitution Definition Let’s look at some

How do you determine a Lim Numerically Graphically Substitution Definition Let’s look at some examples FGuilbert ORRHS

As the number of sides increases, the polygon gets closer to becoming a circle!

As the number of sides increases, the polygon gets closer to becoming a circle! FGuilbert ORRHS

Look at the sequence whose nth term is given by n (n+1) Terms are:

Look at the sequence whose nth term is given by n (n+1) Terms are: 1/2, 2/3, 3/4, 4/5, 5/6 , . . . 99/100, . . . 99999/100000, . They are approaching 1 FGuilbert ORRHS

Look at the sequence whose nth term is given by n (n+1) Terms are:

Look at the sequence whose nth term is given by n (n+1) Terms are: 1/2, 2/3, 3/4, 4/5, 5/6, , , FGuilbert ORRHS

As X approaches Y approaches 0. , As X nears -∞ Y approaches FGuilbert

As X approaches Y approaches 0. , As X nears -∞ Y approaches FGuilbert ORRHS

As X approaches 0 from the l Y approaches -∞. As X nears 0

As X approaches 0 from the l Y approaches -∞. As X nears 0 fro the right, Y approaches ∞. FGuilbert ORRHS

Limits do NOT Agree FGuilbert ORRHS

Limits do NOT Agree FGuilbert ORRHS

Find numerically X f(x) From the left 1. 9 3. 61 As x approaches

Find numerically X f(x) From the left 1. 9 3. 61 As x approaches 1. 99 3. 96 2, f(x) gets 1. 999 3. 996 closer to 4 1. 9999 3. 9996 FGuilbert ORRHS

Find numerically X f(x) From the right 2. 1 4. 41 As x approches

Find numerically X f(x) From the right 2. 1 4. 41 As x approches 2. 01 4. 04 2, f(x) gets 2. 001 4. 004 closer to 4 2. 0001 4. 0004 FGuilbert ORRHS

Find numerically As x approaches 2, from the left f(x) = 4 from the

Find numerically As x approaches 2, from the left f(x) = 4 from the right f(x) = 4 Since both agree, the limit is 4 FGuilbert ORRHS =4

One-sided Limits Limit notation: “The limit of f(x) as x approaches c from the

One-sided Limits Limit notation: “The limit of f(x) as x approaches c from the right side is ” FGuilbert ORRHS

One-sided Limits Limit notation: “The limit of f (x) as x approaches c from

One-sided Limits Limit notation: “The limit of f (x) as x approaches c from the left side is ” FGuilbert ORRHS

If: Then: Limit notation: FGuilbert ORRHS

If: Then: Limit notation: FGuilbert ORRHS

The limit of a function is the value the function approaches, not the actual

The limit of a function is the value the function approaches, not the actual value (if any). 2 not 1 2 FGuilbert ORRHS

2 1 1 2 3 4 At x=1: Does n exist. Left and R

2 1 1 2 3 4 At x=1: Does n exist. Left and R Limits are not = Left limit Right limi FGuilbert ORRHS

Common situations where a limit does not exist (DNE)!! 1) f(x) approaches a different

Common situations where a limit does not exist (DNE)!! 1) f(x) approaches a different number from the right side of c than the left side of c FGuilbert ORRHS

Common situations where a limit does not exist (DNE)!! 2) f(x) increases or decreases

Common situations where a limit does not exist (DNE)!! 2) f(x) increases or decreases without bound as x approaches c. FGuilbert ORRHS

Common situations where a limit does not exist (DNE)!! 3) f(x) oscillates between two

Common situations where a limit does not exist (DNE)!! 3) f(x) oscillates between two fixed values as x approaches c. FGuilbert ORRHS

Once we accept our limit we can go beyond them FGuilbert ORRHS

Once we accept our limit we can go beyond them FGuilbert ORRHS