Jan 28 - Feb 1

Question 1

Limits Laws

Answer 1

Suppose that c is that constant and the limits

lim f(x)
x->a

and

lim g(x)
x->a

1. lim [f(x) + g(x)] = lim f(x) + lim g(x)

all while x approaches a

2. lim [f(x) - g(x)] = lim f(x) - lim g(x)
3. lim [cf(x)] = clim f(x)
4. lim [f(x) g(x)] = lim f(x) * lim g(x)
5. lim f(x)/g(x) = lim f(x)/ li g(x) if lim g(x) DOES NOT equal 0

Question 2

Five Laws Verbally

Answer 2

1. Sum Law: The limit of a sum is the sum of the limits.
2. Difference Law: The limit of a difference is the difference of the limits.
3. Constant Multiple Law: The limit of a constant times a function is the constant times the limit of the function.
4. Product Law: The limit of a product is the product of the limits.
5. Quotient Law: The limit of a quotient is the quotient of the limits.

Question 3

Power Law

Answer 3

(all while x approaches a)

lim [f(x)]^ n = [lim f(x)]^ n

In applying these six limits laws, we need to use two special limits

lim c = c

lim x = a

Question 4

Root Law

Answer 4

(x is approaching a)

lim ^n /f(x) = ^n/limf(x)

assume n is a positive integer

Question 5

Direct Substitution Property

Answer 5

If f is a polynomial or a rational function and a is in the domain of f then

lim f(x) = f(a)
x->a

Question 6

Two-Sided Limit Exists If and Only IF Theorem

Answer 6

lim f(x) = L 
x->a

if and only if

lim f(x) = L = lim f(x)
x->a              x->a+

Question 7

If f(x) /< g(x) when x is near a and the limits of f and g both exist as x approaches a, then

Answer 7

(x is approaching a)

lim f(x) /< lim g(x)

Question 8

Squeeze Theorem

Answer 8

if f(x) /< g(x) /< h(x) when x is near a and

lim f(x) = lim h(x) = L

all while x is approaching a

Question 9

a function f is a continuous at a number a if

Answer 9

(x is approaching a)

lim f(x) = f(a)