Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate form exists, but to find it you must use a technique, such as factor and cancel.

In class: worksheets ( possibilities Worksheet limits or McCafrey) Calculus Date: 9/26/14 Objective: SWBAT define, calculate & apply properties of limits graphically, numerically and now analytically. Do Now – Mini Quiz 5 minutes Take out a piece of paper. Can be a half sheet. HW Requests: HW: pg 30 SM all In class: worksheets ( possibilities Worksheet limits or McCafrey) HW: Complete Worksheets Announcements: Mandatory session Sine and Cosine functions starting with the Unit Circle Quiz Friday To get ahead, You have to do extra!

Show your work then 2. Mini Quiz 7 minutes

Techniques-Finding limits for Rational Expressions Try Substitution, if doesn’t work Try Factor and cancel and then 3. Try Substitution again, if doesn’t work 4. Do DNE or +/- infinity check - If the right and left side limit are not equal the limit does not exist - DNE Let’s go to the SM pg #28 #1-8 (10) HW: pg 30 SM all

If the right and left side limit are not equal the limit does not exist – DNE One sided Limits If the left side number is negative then the lim 𝑥→ 𝑐 − 𝑓 𝑥 =−∞ If the left side number is positive then the lim 𝑥→ 𝑐 − 𝑓 𝑥 =∞ If the right side number is negative then the lim 𝑥→ 𝑐 + 𝑓 𝑥 =−∞ If the left side number is negative then the lim 𝑥→ 𝑐 + 𝑓 𝑥 =∞

Rationalizing Technique If there is a radical in the numerator or the denominator, rationalize, simplify and cancel, then try substitution. Substituting we get Hint: Often you can cancel a common term in the numerator and denominator when simplifying

Rationalizing Technique Rationalize, simplify (cancel) and try substitution. Substituting we get = 1 4

f(0)is undefined; 2 is the limit Try This Find: 2 f(0)is undefined; 2 is the limit

f(0) is defined; 2 is the limit Find: Try This f(0) is defined; 2 is the limit 2 1 1, x = 0

Try This Find the limit if it exists: DNE

Try This Find: if  

Try This Find the limit of f(x) as x approaches 3 where f is defined by:

Example Find the limit if it exists: Try substitution

Example Find the limit if it exists: Substitution doesn’t work…does this mean the limit doesn’t exist?

Use the factor & cancellation technique and are the same except at x=-1

Use the factor & cancellation technique After factoring and cancelling, now try substituting -1 again. = 3

Try This Isn’t that easy? Find the limit if it exists: Did you think calculus was going to be difficult? 5

Try This Solve using limit properties and substitution: 6

Try This Find the limit if it exists:

Example Sometimes limits do not exist. Consider: If substitution gives a constant divided by 0, the limit does not exist (DNE)

Try This Find the limit if it exists: Confirm by graphing The limit doesn’t exist

Lesson Close Name 3 ways a limit may fail to exist.

Exit Ticket In Class: SM – pg 28 #1-5 HW: SM pg 30 #1-15

Try This Find the limit if it exists: -5

Limit properties again The existence or non-existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c. What matters is…what value does f(x) get very, very close to as x gets very, very close to c. This value is the limit.

Watch out for piecewise functions Limits, again! In order for a limit to exist at c, the left-hand limit must equal the right hand limit. If the left-hand limit equals the right hand limit, then the limit exists and we write: Watch out for piecewise functions

When finding the limit of a function it is important to let x approach a from both the right and left. If the same value of L is approached by the function then the limit exist and

Consider Example for and   =?

Try This Graph and find the limit (if it exists): DNE

Example Trig functions may have limits.

Try This

Using the Product Rule Technique  

Important Idea The functions have the same limit as x-1

Try This Graph and on the same axes. What is the difference between these graphs?

Analysis Why is there a “hole” in the graph at x=1?

Example Consider What happens at x=1? Let x get close to 1 from the left: x .75 .9 .99 .999 f(x)

Try This Consider Let x get close to 1 from the right: x 1.25 1.1 1.01 1.001 f(x)

Try This What number does f(x) approach as x approaches 1 from the left and from the right?

Informal Definition of Limit If f(x) becomes arbitrarily close to a single number L as x approaches a number c from either side, the limit of f(x), as x approaches c, is L.

- Definition of Limit Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement means that for each >0 there exists a >0 such that if then .

Basic Limits

Constant Function Limits a and b are both constants This means that for any constant function f(x) = b, as x approaches any constant a, the limit will always be b.

Linear Function Limits The limit of f(x) = x as x approaches any constant is the constant itself.

Exponential Function Limits Just plug in a for x

Properties Let and Scalar multiple: Sum or difference: Product: Quotient: , if K0 Power:

Let's try a practice problem. Property (B) tells us we can split these apart: Using limit (1) and limit (2) from the basic limits, we get:

Putting it all together So, This is called the Substitution method

Try This Solve using limit properties and substitution: 25