Some Ch P and Chapter 1 Review Questions (P2. 69, #79) (P.revw #39) (1.3# 79)(1.4 #84, 93)

 A few problems are explained, including tangent lines, and the Intermediate Value Theorem (IVT).

One Sided, "Type 2" Limits (1.5 #37,49)

 

Type 2 limits are the type that when you substitute the number that x is going towards, you get a fraction with a number in the numerator and a 0 in the denominator (->L/0).  The division by zero is associated with a vertical asymptote, so the limit will go to either positive infinity or negative infinity.

Continuity and the IVT (1.4, #63, 83, 85)

 

Using the requirements of Continuity, we can find a way to make a piece wise function continuous.  We also look at using the Intermediate Value Theorem (IVT) to prove the existence of a zero of a function on a certain closed interval.

Continuity (1.4 #51, #53, and 1.5 #31)

Use limits to test Continuity! We check if the function exits, the limit from left and right exist, and they are all match! 

Continuity and Limits (Section 1.4; 9, 13, 18, 20, 25)

 A function can be continuous at a point, and interval or everywhere.  We can tell if three things are required!

Using a Theorem to find a limit (1.3 #74)

 We can use Theorems together with algebra tricks to find limits that have the indeterminate 0/0 form.

The conjugate comes to the rescue again! (1.3 #75)

To find the limit, we turn to the conjugate once more!

Limits that go to 0/0 (1.3 #53)

Don't fear the "∆x" it works like a single variable, and refers to the change in x.

More Algebra Tricks: The Conjugate (1.3 #52)

 Her we take advantage of (a+b)(a-b) = a² - b ².

to resolve an indeterminate form.

Using Algebra Tricks to Find a Limit (1.3 #51)

 If you substitute 4 for x, we get a 0/0 situation we call an indeterminate form which can resolved with some algebra tricks.