Math 150 Chapter 2 Review

Math 150 Chapter 2 Review (Skip Section 2.4)

1. How many different types of limit are there?

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Can you give an example either in algebra or in geometry to any one of them?

Example. lim f(x) = L

x->a

f(x) = x2. lim x2 = 1

x->1

 

2. How to find a limit?

First thing to do is substitution. Last thing to do is also substitution. If it is a number, +∞, or -∞, it is the limit. If it is ∞ – ∞, 0 ·∞, 0/0, ∞/∞, etc. when we do substitution, we must think other ways to find the limit. Those are the types of indeterminate form. Don’t say the limit does not exist in those cases before we have exhaust all possible ways.

 

3. What is the definition of ‘a function is continuous at a point’?

What kind of function is continued?

What is a discontinuity?

What is a removable discontinuity?

Can you give an example either in algebra or in geometry?

 

Example 1. f(x) = (x2 – 1) / (x – 1). f is discontinuous at x = 1 since f(1) is not defined. But if we define

(x2 – 1) / (x – 1) if x ≠ 1

f(x) = {2 if x = 1

then f(x) is continuous everywhere. x = 1 is a removable discontinuity.

 

Example 2. g(x) = 1 / x. g is discontinuous at x = 0 since lim 1/x does not exist.

x->0

x = 0 is not a removable discontinuity.

4. Any vertical or horizontal asymptote must be a result of a limit.

How to determine a vertical or a horizontal asymptote?

 

5. What is the definition of derivative?

Can you use definition to find the derivative of a function at some point?