Evaluate limits analytically Special limits you need to know

How can we use limits to describe functions behavior graphically or analytically? May 4, 2014 Do Now: Agenda: I. Do Now II. Evaluating limits using Substitution III. Evaluating limits using algebra IV. A little algebra trickery… V. Summary VI. Ticket To Go Take out: pencil, do now packet, notebook Homework: Homework Page ,23-26,37-40 Objectives: Evaluate finite limits graphically, analytically Evaluate right handed and left handed limits graphically, analytically

Sometimes graphing is not the most efficient way to evaluate limits… What other options do I have?

evaluate-limits/ Evaluate limits analytically

First try substitution… 1 st : Always try direct substitution  get answer?? Then you’re done  get ?? Then lim = 0  get ?? Then lim dne (if it can’t be found using one of our other methods)

Why would direct substitution work with these functions? 1) 2)

For a normal function Without holes, jumps, or other weird stuff… JUST PLUG IN. Evaluate limits analytically

Now one more! Now one more!

Let’s try looking at the graph (apparently our graphing function is temporarily back…) Why is this graph linear??? ◦ Can I use some algebra-magic to figure this out?

Now let’s work our algebra-magic!! Now let’s work our algebra-magic!! This accounts for why the graph showed us that the limit equaled -1 and why it’s linear. Why the hole though?? When I can cross something out on top and bottom…I get a HOLE!

One more time now… This function looks so much like our last one…why does this not exist…but the last one did?? When nothing cancels out and we get #/0, then the limit DNE

Conjugate Method Evaluate limits analytically

Finding limits by evaluating one- sided limits

Summary: Flowchart for Evaluating Limits Analytically

Step by Step Approach If attempting to solve a limit analytically: First: Plug in, if it works you’re done. Second: If you get 0/0 try to factor and simplify or conjugate method, then try to plug in again. Third: If you get some number / 0 plug in small values (in a table or in your head) to try and figure out if its ∞, - ∞, or DNE. Fourth: If a fraction or multiplication, find limit of functions separately. Last: Graph if possible to check your answer. Evaluate limits analytically