Unit 1 Flashcards

1
Q

what is a limit?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

point slope form

A

y-y1=m(x-x1)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

when does a limit have an asymptote?

A

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+

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

difference between discontinuity and asymptote algebraically?

A

discontinuity: 0/0

vertical asymptote: #/0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

cases where a limit does not exist

A
  1. f(x) approaches different numbers from the left and right
  2. f(x) approaches -/+ ∞
  3. f(x) oscillates between 2 fixed values
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

5 Limit Laws

A
  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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Properties of Limits

A
  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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What is The Sandwich Theorem?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

3 Special Limits

A

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→∞

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

constant/∞

A

zero

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

∞/constant

A

limit does not exist (infinty)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

7 indeterminant forms

A

1) 0/0

2) ∞/∞

3) ∞-∞

4) 1^∞

5) 0 x ∞

6) 0^∞

7) ∞^0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

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

A

lim f(1/x)
x→0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

what are functions with direct substitution properties called?

A

continuous

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

lim sinx
x→∞

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Definition of Horizontal Asymptote

A

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

17
Q

Definition of a vertical asymptote

A

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+

18
Q

How to find a vertical asymptote?

A

1) x values that are the zeros of the DENOMINATOR

2) constant/ zero

19
Q

How to determine removable points (holes) ?

A

1) x values that are zeros of NUMERATOR AND DENOMINATOR

2) 0/0

20
Q

How to find a horizontal asymptote?

A

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

2) numerator and denominator have same degree

3) numerator degree < denominator degree

21
Q

How to find slant asympotote?

A

1) numerator degree is exactly one more than denominator degree

2) divide and ignore remainder

22
Q

3 requirements for a function to be continuous

A
  1. f(a) is defined
  2. lim f(x) exist
    x→a
  3. lim f(x) = f(a)
    x→a
23
Q

what are the two different types of discontinutiy?

A

removable and nonremovable

24
Q

when is a discontinuity removable?

A
  1. when f can be redefined so that f is continuous
  2. hole
  3. 0/0
25
Q

when is a discontinuity nonremovable?

A
  1. when f cannot be redefined so that f is continuous
  2. asymptotes, jumps, limits dne
  3. # /0
26
Q

how to check for continuity?

A
  1. check function value
  2. check limit value (both left and right)
  3. Function V =Limit Value = continuity
27
Q

requirements to use The Intermediate Value Theorem

A
  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

28
Q

what does the IVT say?

A

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

29
Q

formula for average rate of change

A

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

30
Q

formula for instantaneous rate of change/ slope of tangent line

A

f’(x)= lim f(x+h) - f(x) / h
h-> 0

31
Q

velocity formula

A

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