Unit 1

Question 1

what is a limit?

Answer 1

the limit, L, of f(x) exists if f(x) becomes close to the same, single, real number L as x approaches c from both the right and left sides of c

Question 2

point slope form

Answer 2

y-y1=m(x-x1)

Question 3

when does a limit have an asymptote?

Answer 3

lim f(x)=∞ lim f(x)=∞ lim f(x)=∞ lim f(x)=-∞ lim f(x)=-∞ lim f(x)=-∞ x→a x→a- x→a+ x→a x→a- x→a+

Question 4

difference between discontinuity and asymptote algebraically?

Answer 4

discontinuity: 0/0

vertical asymptote: #/0

Question 5

cases where a limit does not exist

Answer 5

1. f(x) approaches different numbers from the left and right
2. f(x) approaches -/+ ∞
3. f(x) oscillates between 2 fixed values

Question 6

5 Limit Laws

Answer 6

1. lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
   x→a x→a x→a
2. lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
   x→a x→a x→a
3. lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
   x→a x→a x→a
4. lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
   x→a x→a x→a
5. lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
   x→a x→a x→a

Question 7

Properties of Limits

Answer 7

1. The limit of a sum is the sum of their limits
2. The limit of a difference is the difference of the limits
3. The limit of a constant times a function is the constant times the limit of the function
4. The limit of a product is the product of the limits
5. The limit of quotient is the quotient of the limits (as long as the denominator is not 0)
6. The limit of a function raised to a power is the limit raised to that same power

Question 8

What is The Sandwich Theorem?

Answer 8

If the h(x)≤f(x)≤g(x) for all x in an open interval containing c, except c itself, and if lim h(x)=L=lim g(x),
x→c x→c
then, lim f(x) exists and is equal to L
x→c

Question 9

3 Special Limits

Answer 9

1) lim sinx/x = 1 & lim x/sinx = 1
x→0 x→0

2) lim 1-cosx/x = 0
x→0

3) lim (1+x)^1/x = e & lim (1+1/x)^x = e
x→0 x→∞

Question 10

constant/∞

Answer 10

zero

Question 11

∞/constant

Answer 11

limit does not exist (infinty)

Question 12

7 indeterminant forms

Answer 12

1) 0/0

2) ∞/∞

3) ∞-∞

4) 1^∞

5) 0 x ∞

6) 0^∞

7) ∞^0

Question 13

How can we rewrite the limit : lim f(x)
x→∞

Answer 13

lim f(1/x)
x→0

Question 14

what are functions with direct substitution properties called?

Answer 14

continuous

Question 15

lim sinx
x→∞

Answer 15

dne as the values of sinx oscillate between 1 and -1

Question 16

Definition of Horizontal Asymptote

Answer 16

The line y=b is a HA if either lim f(x) = b or lim f(x) = b
x→∞ x→-∞

Question 17

Definition of a vertical asymptote

Answer 17

f(x) has a VA at x=a if

lim f(x)=∞ lim f(x)=∞ lim f(x)=∞ lim f(x)=-∞ lim f(x)=-∞ lim f(x)=-∞ x→a x→a- x→a+ x→a x→a- x→a+

Question 18

How to find a vertical asymptote?

Answer 18

1) x values that are the zeros of the DENOMINATOR

2) constant/ zero

Question 19

How to determine removable points (holes) ?

Answer 19

1) x values that are zeros of NUMERATOR AND DENOMINATOR

2) 0/0

Question 20

How to find a horizontal asymptote?

Answer 20

1) when f(x) approaches a limit, L, as x approached +/- ∞

2) numerator and denominator have same degree

3) numerator degree < denominator degree

Question 21

How to find slant asympotote?

Answer 21

1) numerator degree is exactly one more than denominator degree

2) divide and ignore remainder

Question 22

3 requirements for a function to be continuous

Answer 22

1. f(a) is defined
2. lim f(x) exist
   x→a
3. lim f(x) = f(a)
   x→a

Question 23

what are the two different types of discontinutiy?

Answer 23

removable and nonremovable

Question 24

when is a discontinuity removable?

Answer 24

1. when f can be redefined so that f is continuous
2. hole
3. 0/0

Question 25

when is a discontinuity nonremovable?

Answer 25

1. when f cannot be redefined so that f is continuous
2. asymptotes, jumps, limits dne
3. # /0

Question 26

how to check for continuity?

Answer 26

1. check function value
2. check limit value (both left and right)
3. Function V =Limit Value = continuity

Question 27

requirements to use The Intermediate Value Theorem

Answer 27

1. f is continuous on [a,b]
2. m is any number between f(a) and f(b)

*must state that function is continuous always

Question 28

what does the IVT say?

Answer 28

If f is continuous , [a,b], and m is any number between f(a) and f(b), then there is at least one number c between a and b that gives f(c)=m

Question 29

formula for average rate of change

Answer 29

f(x+h) - f(x) / h

Question 30

formula for instantaneous rate of change/ slope of tangent line

Answer 30

f’(x)= lim f(x+h) - f(x) / h
h-> 0

Question 31

velocity formula

Answer 31

v(t) = s’(t) = lim s(x+h) - s(x) / h
h-> 0