16.1 Limit of a Function of a Real Variable

(Don’t write – just explore) Let’s look at a graph and look at its table of values. Desmos…. In box 1, type f (x) = 0.5x + 3 In box 2, go to & choose table In y spot, delete y and type f (x) 1 is already there, let’s do some more (type in x column) 1.5 1.75 1.85 1.95 1.999 2.001 2.05 2.15 2.25 Now, look at the graph & the table of values As the x-values get closer to 2, what do the y-values get close to? So… as x approaches 2, y approaches 4 4 *this idea has a name…. but something to watch first…

Why does she say “the limit does not exist?” Back to Desmos…. Delete equation in box 1 and type Look at this graph & the table of values Answer the same question as before… As the x-values get closer to 2, what do the y-values get close to? Discuss with a partner It depends on if you are approaching 2 from the left or the right From the left  y-values get very large From the right  y-values get very small So, “the limit does not exist.” Let’s ask another question … As the x-values get SUPER BIG or SUPER SMALL, what do the y-values get close to? 2

These ideas of the x-values getting close to something and the resulting y-values getting close to something is the idea behind a LIMIT. (Now write) Notation: Ex 1) Using the graph (on Desmos) of each function, find the limit. gets way big gets way small

Sometimes as x approaches a specific x-value from the right (means traveling left) or from the left (means traveling right), the y-values do not match – that is ok! (from the right) (from the left) If they are the same, , we say the limit exists at a and (*Special note: the function may or may not have the same value as the limit at a particular a – it’s ok!)

Ex 2) the limit exists! If you are dealing with a polynomial function (like we did at the beginning of the lesson), the value of the limit IS the value of the function. If a function f (x) is a polynomial function on an interval (a, b) and c is a real number on that interval, then

Ex 3) Find the limit of each function. (plug it in) 14 2 We may also have to examine a graph to find a limit. Ex 4) at 0, this function is undefined, but as it approaches 0 from the right and the left, it approaches the y-value of 1 *graph on Desmos

A precise formal mathematical definition of a limit is as follows:

Homework #1601 Pg 857 #1, 3, 5, 7–14, 17, 19, 21, 23, 25, 26, 30, 32–36, 38, 41–44