Express the repeating decimal 0.5757... as the ratio of two integers without your calculator.

Warm-Up Express the repeating decimal 0.5757... as the ratio of two integers without your calculator.

Warm-=Up x = 0.57 Express the repeating decimal 0.5757... as the ratio of two integers without your calculator. 100x = 57.57 99x = 57 x = 57/99

What is Calculus? There are only 3 main concepts in calculus. The Limit 2) The Derivative The Integral

What is Calculus? There are only 3 main concepts in calculus. The Limit 2) The Derivative The Integral You will need a graphing calculator.

1-2:Finding Limits Graphically and Numerically Objectives: • Understand the concept of a limit • Calculate limits

Important Ideas • Limits are what make calculus different from algebra and trigonometry • Limits are fundamental to the study of calculus • Limits are related to rate of change • Rate of change is important in engineering & technology

m=3 m=2 m=1 m=-1 • Slope is a rate of change Analysis f(x) • Rate of change is constant at every value on a linear f(x) m=2 x

Analysis • Rate of change is different at every value on a non-linear f(x) f(x) • Rate of change is the slope of the tangent line at a point x

Important Ideas • The slope of a secant line is an average rate of change • The slope of a tangent line is an instantaneous rate of change at a point

We don’t know how to calculate instantaneous rate of change ,therefore, Analysis • We know how to calculate average rate of change • The tangent line problem… Go to Sketchpad

Warm-Up-You need a graphing calculator. I’m using a TI-84. Put your signature pages in the box

Important Idea f(x) InstantaneousRate of change is different at every point on f(x) x Limits are used to calculate the slopes of the tangents

Example • Graph: 2. Trace to x=2. 3. Zoom in at least 4 times. 4. Describe the graph.

Example Consider What happens at x=1? Let x get close to 1 from the left:

Try This Consider Let x get close to 1 from the right:

Try This What number does f(x) approachasx approaches 1 from the left and from the right?

Try This Graph and on the same axes. What is the difference between these graphs?

Analysis Why is there a “hole” in the graph at x=1?

Consider =? Example for and for x=1

Try This Find: if

Important Idea The existence or non-existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c.

Important Idea What matters is…what value does f(x) get very, very close to as x gets very,veryclose to c. This value is the limit.

Try This Find: 2 f(0)is undefined; 2 is the limit

Find: 2 1 f(0) is defined; 2 is the limit Try This

Warm-Up

Try This Find the limit of f(x) as x approaches 3 where f is defined by:

Example Graph and find the limit (if it exists):

Important Idea Some limits do not exist. If f(x) approaches as x approaches c, we say that the limit does not exist at c or, sometimes we saythe AP Exam says the limit approaches infinity at c.

Example Find the limit if it exists:

But… Does not exist Important Idea

Definition If a function has a limit, the limit from the right must equal the limit from the left.

Example 1.Graph using a friendly window: 2. Zoom at x=0 3. Wassup at x=0?

Important Idea If f(x) bounces from one value to another (oscillates) as x approachs c, the limit of f(x) does not exist at c:

Lesson Close Name 3 ways a limit may fail to exist.

Assignment Page 54 Problems 1 - 7 odd, 8 – 24 all In class, we will not cover the formal definition of a limit, sometimes called epsilon-delta definition. I’ll talk about it in NMSI tutoring.

Express the repeating decimal 0.5757... as the ratio of two integers without your calculator.