chapter 2 (limits) Flashcards
v average
x2-x1/y2-y1
right sided limits
what x approaches as it approaches from the right
left sided limits
what x approaches fro teh left
first way to attempt to take a limit
plug it in
lim (a+b)
lim a +lim b
lim a-b
lim a - lim b
lim ca
a lim c
lim (ab)
(lim a)(lim b)
lim (a/b)
lim a/lim b
lim a^n
(lim a)^n
lim a^(a/b)
(lim a)^a/b
squeeze theorem
f
vertical asymptotes
when the denominator +0
hole
when the numerator and denominator cancel out
horizontal asymptotes
if denominator degree is larger than numerator then asymptote is 0
if they equal it is the fraction of there coefficients
if degree of numerator is larger then no HA
slant asymptotes
occur when the degree of the numerator is larger than the denominator
polynomial long devision
- write the denominator outside the division symbol and the numerator in it
- ask what multiplies by the first value in the denom to get the first value of the numerator
- subtract after this
- with the new line ask the same thing again
- cancel until you get the remainder. you dont need it for slant asymptotes.
lim (x to inf) e^x
inf
lim (x to -inf) e^x
0
lim (x to inf) e^-x
0
lim (x to -inf) e^-x
inf
lim (x to inf) ln x
inf
lim (x to 0+)
-inf
for a point to be continuous what must be true
it must be defined
limit must exist
value of f(a) = limit at that point
a rational function is continuous for all values P/q wehn
q does not equal 0
if f is cont and g is cont then F(g(x))
is cont
if g is cont at a and f is cont at g(a)
then the limit is lim(x to a) is f(lim g(a))
if a function is continuous on and interval then the inverse is
also continuius
intermediate value theorem
f is continuous from a to b and L is a number in between f(a) and f(b) then there exists one number c (between a and c) in that f(c) is L
how to find epsilon given delta is [x-a]<1
and {fx-L]
look at graph. find the values vertically form a+1 and a-1
then find the horizontal equivalent
the difference from the horizontal number at L
how to find delta given epsilon is {fx-L]<1
and [x-a]
look at graph. find the values horizontalv form L+1 and L-1
then find the vertically equivalent
the difference from a to its extremeties is delta
how to do a limit with the formal definition
ex prove that lim (4x-15) =1
- set as [f(x)-L]
