Limits Numerically WarmUp What do you think the

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. The height of the line y=2 is always 2, so the “intended height” or “where it is heading towards” is always going to be 2!!

Objectives n To determine when a limit exists. n 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 The limit, as x approaches 2, of f (x) is 4. or The limit of f (x), as x approaches 2, is 4.

Video Clip from Calculus-Help. com When Does a Limit Exist?

When does a limit exist? n. 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… n 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? n. A limit will not exist if there is a break in the graph of a function. n If the height arrived at from the left does not match the height arrived at from the right, then the limit does not exist. n 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 n Read as: “The limit of f (x) as x approaches 4 from the right equals 2. ” n This means x approaches 4 with values greater than 4.

Left-hand Limit: the height arrived at from the left n Read as: “The limit of f (x) as x approaches 4 from the left equals 1. ” n This means x approaches 4 with values less than 4.

General Limit n. A general limit exists on f (x) when x = c, if the left- and right-hand limits are both equal there. Mathematic Notation: In other words: f (x) L as x c

Finding Limits =7 =7 x g (x) . 9 6. 71 . 99 6. 9701 . 999 6. 997 If a function approaches the same value from both directions, then that value is the limit of the function at that point. =7 x g (x) 1. 1 7. 31 1. 01 7. 0301 1. 001 7. 003

Finding Limits = DNE or NL =3 x h (x) – 1. 1 3. 1 – 1. 01 3. 01 – 1. 001 3. 001 = – 3 If the Left-hand limit and the Right-hand limit are not equal, the general limit does not exist. x h (x) –. 9 – 2. 9 –. 99 – 2. 99 –. 999 – 2. 999

Finding Limits = DNE or NL = NL x j (x) 2. 9 – 44. 1 2. 99 – 494 2. 999 – 4994 If either the Left-hand limit, Right-hand limit, or both do not exist, the general limit will not exist. = NL x j (x) 3. 1 56. 1 3. 01 506. 01 3. 001 5006

Conclusion n A limit is the intended height of a function. n A limit will exist only when the left- and righthand limits are equal. n A limit can exist if there is a hole in the graph. n A limit will not exist if there is a break in the graph.