Unit 1: Limits and Continuity

Limits Overview
Limits Overview
Limit Notation (Left-Hand, Right-Hand, Overall)
Limit Notation
3 Topics
Left Hand Limit (LHL)
Right Hand Limit (RHL)
Overall Limit
Finding Limits Using a Graph
Finding Limits Using a Graph
5 Topics
Method: Finding Limits of a Graph
Example 1: Left Hand Limit
Example 2: Right Hand Limit
Example 3: Overall Limit Exists
Example 4: Overall Limit Does Not Exist
Finding a Limit Using Equations
Finding Limits Using an Equation
Option 1: Plug it in!
3 Topics
Identifier: Limit Option 1
Method: Limit Option 1
Example 1: Option 1 (Plug it in)
Option 2: Factor and Cancel
4 Topics
Identifier: Factor and Cancel
Method: Factor and Cancel
Example 1: Factor and Cancel
Example 2: Factor and Cancel
Option 3: Complex Fractions
3 Topics
Identifier: Complex Fraction Limits
Method: Complex Fraction Limits
Example 1: Complex Fraction Limit
Option 4: Square Roots
3 Topics
Identifier: Square Root Limits
Method: Square Root Limits
Example 1: Square Root Limits
Option 5: Absolute Value
3 Topics
Identifier: Absolute Value Limits
Method: Absolute Value Limits
Example 1: Absolute Value Limit
Option 6
Trig Functions Category 1: Simplify then Plug it in
4 Topics
Identifier: Trig Function Limits Category 1
Method: Trig Function Limits Category 1
Example 1: Trig Function Limit Category 1
Example 2: Trig Function Limit Category 1
Option 7
Trig Functions Category 2: Sine Limit Identity
3 Topics
Identifier: Trig Function Category 2
Method: Trig Function Limits Category 2
Example 1: Trig Function Limit Category 2
Option 8
Trig Functions Category 3: Cosine Limit Identity
3 Topics
Identifier: Trig Function Category 3
Method: Trig Function Category 3
Example 1: Trig Function Limit Category 3
Option 9: Vertical Asymptotes (Limits that equal infinity)
6 Topics
Identifier: Vertical Asymptotes
Method: Vertical Asymptotes
Example 1: Vertical Asymptotes
Example 2: Vertical Asymptotes
Example 3: Vertical Asymptotes
Example 4: Vertical Asymptotes
Option 10: Horizontal Asymptotes (Limits whose x-value is heading towards infinity)
10 Topics
Identifier: Horizontal Asymptotes
Method 1: Leading Behavior
Method 2: Dividing by the Highest Power of x
Example 1: Leading Behavior (Top-Heavy)
Example 2: Leading Behavior (Bottom-Heavy)
Example 3: Leading Behavior (Balanced)
Example 4: Dividing by the Highest Power of x (Top-Heavy)
Example 5: Dividing by the Highest Power of x (Bottom-Heavy)
Example 6: Dividing by the Highest Power of x (Balanced)
Example 7: 2 Different Horizontal Asymptotes (Pro Level)
Option 11: Squeeze Theorem (Sandwich Theorem)
3 Topics
Identifier: Squeeze Theorem
Method: Squeeze Theorem
Example 1: Squeeze Theorem
Continuity
Definition of Continuous Function
Determining Continuity: Given a Graph
3 Topics
Identifier: Continuity Using a Graph
Method: Continuity Using a Graph
Example 1: Continuity Using a Graph
Determining Continuity: Given an Equation
3 Topics
Identifier: Continuity Using an Equation
Method: Continuity Using an Equation
Example 1: Continuity Using an Equation
Intermediate Value Theorem
3 Topics
Identifier: Intermediate Value Theorem
Method: Intermediate Value Theorem
Example 1: Intermediate Value Theorem
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Method: Absolute Value Limits

Unit 1: Limits and Continuity Option 5: Absolute Value Method: Absolute Value Limits

Divide and Conquer

Step 1: Try Option 1: Plug it in

Step 2: Divide the absolute value equation into two equations; left and right.

In order to do algebra on an absolute value equation, you need to split the equation up into its two component pieces. Remember every absolute value equation is really made up of two different equations. One equation coming in from the left,the negative side, and another equation coming in from the right, the positive side.

 

–           To create the left equation, you will drop the absolute value bars and replace them with parenthesis and a negative time in front.

Left Example: | x + 2 | = – ( x + 2 )

–           To create the right equation, you will drop just drop the absolute value bars.

Right Example: | x + 2 | = x + 2

Step 3: Determine the overall limit by finding your left-hand limit and right-hand limit using the equations you created in Step 2 .

Finding these limits will require you to use some of your other limit options. Usually Option 1: Plug it in and Option 2: Factor and Cancel.

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