derivatives lesson oct 15
TRANSCRIPT
LIMITS
We say that the limit of f(x), as x approaches a written
limx a
f (x) = L
if we can take outputs f(x) as close to L as we wish by taking inputs x sufficiently close to a but not equal to a.
Graph the function f(x) = x2 9x 3
Now lets estimate limx 3
f(x)
On graphing calculator press 2nd [Tblset]
Then press 2nd [TABLE]
Graph f(x) = sinxx
estimate limx 0
f(x)
Graph f(x) = xx
limx 0+ x
x= 1 lim
x 0 xx
= 1
Right hand limit: limx a+
f(x) = L means that f(x) can be
made as close to L as we wish by taking x sufficiently close to a but greater than a
x alim f(x) = L means that f(x) can be
made as close to L as we wish by taking x sufficiently close to a but less than a
Left hand limit:
sinxx
limx 0+
sinxx
limx 0
sinxx
limx 0
= = = 1
limx 0 x
xlimx 0+ x
x limx 0 x
x= therefore, does not exist
Rules for calculation limits
1. Sum Rule: x alim [ f(x) + g(x) ] =
x alim f(x) +
x alim g(x) = L + M
L M
[ f(x) g(x) ]2. Difference Rule:x alim =
x alim f(x)
x alim g(x)
=
f(x) =x alim g(x)3. Product Rule: x a
limx alim[ f(x) g(x) ] = LM..
4. Constant Multiple Rule:x alim f(x)k = kL
5. Quotient Rule:x alim =
f(x)g(x)
x alim f(x)
x alim g(x)
= LM
=if M 0
Findx 2lim x2
3x + 4
x 2lim x2
3x + 4= x 2
lim x2
x 2lim (3x + 4)
= x 2lim x
x 2lim x
x 2lim
x 2limx3 4+
2
5
Find x 2lim x2 5x + 6
x 2
Questions 1 19 odd only
Exercise 2.7