Limits Overview
Limit Notation (Left-Hand, Right-Hand, Overall)
Finding Limits Using a Graph
Finding a Limit Using Equations
Continuity

Example 1: Absolute Value Limit

lim y 1 y 2 1 |2 y + 2|

Step 1: Try Option 1: Plug it in

Always try plugging in the x value you are heading towards.

In this example we get division by zero, which means Option 1 has failed.

lim y 1 ( 1 2) 1 |2 ( 1) + 2| = 0 0

Step 2: Split the equation into its two equations. A left-hand limit and a right-hand limit.

lim y 1 y 2 1 |2 y + 2|

LHL: Drop the absolute value bars and replace them with a (          ) .

lim y 1 y 2 1 ( 2 y + 2 ) =

RHL: Drop the absolute value bars

 

lim y 1 + y 2 1 2 y + 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.

 

To solve both the LHL and the RHL we will use two previous limit options. First, we will use Option 2: Factor and Cancel, and then back to Option 1: Plug it in.

lim y 1 y 2 1 ( 2 y + 2 ) =

lim y 1 ( y 1 ) ( y + 1 ) ( 2 y + 2 ) =

lim y 1 ( y 1 ) ( y + 1 ) 2 ( y + 1 ) =

lim y 1 ( y 1 ) ( y + 1 ) 2 ( y + 1 ) =

lim y 1 ( y 1 ) 2 =

lim y 1 (y 1) 2 = ( 1) 1 2 = ( 2) 2 = 1

 

lim y 1 + y 2 1 2 y + 2 =

lim y 1 + ( y 1 ) ( y + 1 ) 2 y + 2 =

lim y 1 + ( y 1 ) ( y + 1 ) 2 ( y + 1 ) =

lim y 1 + ( y 1 ) ( y + 1 ) 2 ( y + 1 ) =

lim y 1 + ( y 1 ) 2 =

 

lim y 1 + ( y 1 ) 2 = ( 1) 1 2 = ( 2) 2 = 1

 

 

LHL RHL

1 1

lim y 1 y 2 1 |2 y + 2| = Does Not Exist

Final Result:

lim y 1 y 2 1 |2 y + 2| = Does Not Exist

 

Meaning:

The overall limit as y approaches 1 of y 2 1 |2 y + 2| Does Not Exist.

 

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