Limits Graphically & Limit Properties Day 1 Limits Graphically & Limit Properties

What is a limit? A limit describes how the output values of a function behave as input values approach some given #, “c” Notation: Read as “limit of f(x) as x approaches c is equal to L”

3 Kinds of limits THE Limit (double-sided limit) Left-hand limit Limit of f(x) as x approaches c from either direction. Only exists if left-hand and right-hand limits are the same. Left-hand limit Limit of f(x) as x approaches c from the left side. Right-hand limit Limit of f(x) as x approaches c from the right side.

Common Misconception #1 A function does not have to be defined at “c” in order for the limit to exist.

Common Misconception #2 If a function is defined at “c”, f(c) does not necessarily have to equal L.

Practice

Practice

Practice

Practice 10

Draw a graph such that

Draw a graph such that

Draw a graph such that

Draw a graph such that

Draw a graph such that

Draw a graph such that

Two Cases for When the Limit is D.N.E. (Does Not Exist) Behavior differs from the left and right Oscillating Behavior Ex/

Increase or decrease without bound Unbounded Behavior Increase or decrease without bound Can you think of a parent function that would fall in this category? More descriptive than DNE

Some Basic Limits The limit of a constant function is the constant.

The limit at any x-value on the line y=x IS the x-value itself.

One more: The limit at any x-value of a function f(x)=xn (where n is an integer) is the x-value raised to the nth power.

Practice:

Theorem: Properties of limits

Theorem: Properties of limit The limit of a sum is equal to the sum of the limits The limit of a difference is equal to the difference of the limits

Theorem: Properties of limit The limit of a product is equal to the product of the limits

Theorem: Properties of limit The limit of a quotient is equal to the quotient of the limits

Theorem: Properties of limit