Warm Up: No Calc 1. Find all asymptotes for (A) x=1, x=-1, y=1 (B) x=1, y=1(C) x=1, x=-1, y=0 (D) x=1, x=-1(E) y=1 2. 3. Use properties of logarithms.

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EVALUATING LIMITS ANALYTICALLY

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Presentation transcript:

Warm Up: No Calc 1. Find all asymptotes for (A) x=1, x=-1, y=1 (B) x=1, y=1(C) x=1, x=-1, y=0 (D) x=1, x=-1(E) y= Use properties of logarithms to decide which of the following is largest. Pick up new packet!

If we increase the number of sides of the polygon, what can you say about the polygon with respect to the circle?

What is a limit? Limit is the value of Y as X approaches a given #:

3 Kinds of Limits: Left – Hand Limit As x approaches from the left side of c Right – Hand Limit As x approaches from the right side of c. Limit (double – sided) As X approaches c from either direction. Only exists if left- hand and right-hand are the same.

When do limits not exist? DNE Video

THM: Existence of a Limit

Example 1: Find the following limits

Practice

11

00

-2 

Grab a graphing board, marker, and towel

Limit Properties These are important!

Limits to Know Let b & c be real numbers and let n be a positive integer. 1. The limit of a constant function is the constant. 2. The limit at any x-value on the line y=x IS the x-value itself. 3. The limit at any x-value of any function of the form y = x n is the x-value raised to the nth power.

Practice:

Properties of Limits Let b & c be real # and n a positive int. and Scalar multiple Sum or Differ. Product Quotient Power

Practice 1. 2.

Another nice thing about limits… They help us find holes in the graph. Ex: What will happen at x=1?

x f(x)?

While f(1) D.N.E., as x moves arbitrarily close to 1 from the left and right, f(x) moves arbitrarily close to 3. “The limit of f(x) as x approaches 1 is 3”

Example: Graph

Homework: pg. 65 (1 – 4, 37 – 48, 79-82) Packet pg. 1