Warm-Up: What do you think the following limit equals? If you are unsure at least recall what a limit is and see if that helps direct you

Limits Numerically

• Warm-Up: What do you think the following limit equals? If you are unsure at least recall what a limit is and see if that helps direct you.

Objectives

• To determine when a limit exists.

• To find limits using a graphing calculator and table of values.

• TS: Explicitly assessing information and drawing conclusions.

What is a limit?

• A limit is the intended height of a function.

How do you determine a function’s height?

• Plug an x-value into the function to see how high it will be.

Can a limit exist if there is a hole in the graph of a function?

• Yes, a limit can exist if the ultimate destination is a hole in the graph.

Limit Notation

Video Clip from Calculus-Help.com

• When Does a Limit Exist?

When does a limit exist?

• A limit exists if you travel along a function from the left side and from the right side toward some specific value of x, and…

• As long as that function meets in the middle, as long as the heights from the left AND the right are the same, then the limit exists.

When does a limit not exist?

• A limit will not exist if there is a break in the graph of a function.

• If the height arrived at from the left does not match the height arrived at from the right, then the limit does not exist.

• Key Point: If a graph does not break at a given x-value, a limit exists there.

One Sided Limits

Right-hand Limit: the height arrived at from the right

• Read as: “The limit of f (x) as x approaches 4 from the right equals 2.”

• This means x approaches 4 with values greater than 4.

Left-hand Limit: the height arrived at from the left

• Read as: “The limit of f (x) as x approaches 4 from the left equals 1.”

• This means x approaches 4 with values less than 4.

General Limit

• A general limit exists on f (x) when x = c, if the left- and right-hand limits are both equal there.

Finding Limits

Finding Limits

Finding Limits

Conclusion

• A limit is the intended height of a function.

• A limit will exist only when the left- and right-hand limits are equal.

• A limit can exist if there is a hole in the graph.

• A limit will not exist if there is a break in the graph.