Unit 1 Flashcards
what is a limit?
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
point slope form
y-y1=m(x-x1)
when does a limit have an asymptote?
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+
difference between discontinuity and asymptote algebraically?
discontinuity: 0/0
vertical asymptote: #/0
cases where a limit does not exist
- f(x) approaches different numbers from the left and right
- f(x) approaches -/+ ∞
- f(x) oscillates between 2 fixed values
5 Limit Laws
- lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
x→a x→a x→a - lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
x→a x→a x→a - lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
x→a x→a x→a - lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
x→a x→a x→a - lim(f(x)+g(x)) = limf(x)+limg(x)=L+M
x→a x→a x→a
Properties of Limits
- The limit of a sum is the sum of their limits
- The limit of a difference is the difference of the limits
- The limit of a constant times a function is the constant times the limit of the function
- The limit of a product is the product of the limits
- The limit of quotient is the quotient of the limits (as long as the denominator is not 0)
- The limit of a function raised to a power is the limit raised to that same power
What is The Sandwich Theorem?
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
3 Special Limits
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→∞
constant/∞
zero
∞/constant
limit does not exist (infinty)
7 indeterminant forms
1) 0/0
2) ∞/∞
3) ∞-∞
4) 1^∞
5) 0 x ∞
6) 0^∞
7) ∞^0
How can we rewrite the limit : lim f(x)
x→∞
lim f(1/x)
x→0
what are functions with direct substitution properties called?
continuous
lim sinx
x→∞
dne as the values of sinx oscillate between 1 and -1
Definition of Horizontal Asymptote
The line y=b is a HA if either lim f(x) = b or lim f(x) = b
x→∞ x→-∞
Definition of a vertical asymptote
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+
How to find a vertical asymptote?
1) x values that are the zeros of the DENOMINATOR
2) constant/ zero
How to determine removable points (holes) ?
1) x values that are zeros of NUMERATOR AND DENOMINATOR
2) 0/0
How to find a horizontal asymptote?
1) when f(x) approaches a limit, L, as x approached +/- ∞
2) numerator and denominator have same degree
3) numerator degree < denominator degree
How to find slant asympotote?
1) numerator degree is exactly one more than denominator degree
2) divide and ignore remainder
3 requirements for a function to be continuous
- f(a) is defined
- lim f(x) exist
x→a - lim f(x) = f(a)
x→a
what are the two different types of discontinutiy?
removable and nonremovable
when is a discontinuity removable?
- when f can be redefined so that f is continuous
- hole
- 0/0
when is a discontinuity nonremovable?
- when f cannot be redefined so that f is continuous
- asymptotes, jumps, limits dne
- # /0
how to check for continuity?
- check function value
- check limit value (both left and right)
- Function V =Limit Value = continuity
requirements to use The Intermediate Value Theorem
- f is continuous on [a,b]
- m is any number between f(a) and f(b)
*must state that function is continuous always
what does the IVT say?
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
formula for average rate of change
f(x+h) - f(x) / h
formula for instantaneous rate of change/ slope of tangent line
f’(x)= lim f(x+h) - f(x) / h
h-> 0
velocity formula
v(t) = s’(t) = lim s(x+h) - s(x) / h
h-> 0
