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Class notes 201-NYA-05
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These notes cover all topics in Calculus 1
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Limits- value function approaches as the input approaches some value
that a
lim
1 a function f has a limit Las X approaches a , written xeaf(x) = L if the
close to the number
Las we please by taking x
sufficiently close to a lon eit
limit definition : lim f(x+h) f(x) -
neo
h
ex)
im 4x2 + lim2x +im 2
X-2 X-2
=4 [limx] + 2 limx+ lim 2
X+ 2
X92Xe2
=
4[232 + 2(2) + 2
=
16 + 4 + 2 =
22 : The limit exists & converges to 22
limits
Evaluating ~ ratio of 2 p
· substitution property : if f(x) is a polynomial or a rational function
< then lim f(x) F(a) lother functions benefit this property)
may
: =
Xe A
< lim DN
& of limits 1)
2
types : xea
limits can
equal a number ,
infinity or
m
2)x > 10
/im ==DNE ex3) lim
ex1) ex 2) lim -
X = 0 = DNE
10gaX
X- 0
"
& DNE
" -
DNE ⑧
-
&
!
E
b
[ 2
b
[
1 both sides aren't equal
Evaluating limit techniques
C First case : lim P(x) & the limit will evaluate to 0/0 when .
X=a Then ,
x+
aQ(x)
1) if P & Q are
polynomials & P(a)=Q(a) = 0 , then X-a is a factor of bot
<lim P(x) (a)(g(x)) g(x) division if ca
xaQ(x)
= =
(might have to use long
(a)(k(x))k(X)
2) if P & Q are not polynomials & one or both has a radical , we rationaliz
3) if P & Q are not polynomials & contains a fraction on either the
, =lim x2 3x -1 (x + 2) x2 + 2x (x -1)(x + 3) (x + 3)) =
+ 3
( 4
-
-
= = =
-
-
Xe1- -
(x -
1) -
(X -
1) -
(x1) : the lim D
X-
= lim x2 + 3x -1 -
(x + 2) =
x2 + 2x -
3 =
(x)(x + 3) = (x + 3) = 4
X- It (X 1) - X -
1 -1
ex 3) lim X -
2x-1 .
x + 2x -
1 .
x+3 + 2
x+1x + 3
-
2 X+ 2x -
1 X+ 3 +2
= lim (x= 2x + 1) .
(x + 3 + 2)
X-1
X+ 3 -
4 X + 2x -
1
= lim (x-1)((x) .
X+ 3 + 2
X- /
(1) X+ 2X -
1
= lim(x -1)(x + 3 + 2) = 0
X91 X+ 2X - 1
Limits
here is a problem using long division :
ex : lim x3 + 6x2 + 4x-1
Xe -
It x2 + 2x + 1
x2 + 5x -
1
lim x3 + 6x2 + 4x 1
X + 1x3 + 6x2 +
=
-
> 4X 1
It
-
Xe
(X + 1)2
-
- x3 + x2 d
=lim(xF1)(y2 + 5x 1) -
05x2 + 4x
X- -
It
+ 1)2
(x- 5x2 + 5xx
-
0 -
X -
1
lim
5
= =
-
0(V A
--X 1
-
= .
-
Xe -
It
O
Vertical Asymptotes
& the line x=a is called a V A .
.
of the curve f(x) if at least one of the
followi
f(x) f(x) = f(x)
1)
lim
= 3)
lim 5) lim d
=
Xe at
2) lim 4) f(x) = f(x)
f(x) = d
lim 6) lim -
-
=
X- a xe at
:
only one of the above conditions must be met for there to be a V A
. .
at
( sometimes ,
V A . . occur when the limit evaluates to
& we've already seen functions with V A .
:
im logaX=-D
, a clim1
Xe0X2
= @ (both sides go to
infinity
, =
Xim 3
+ ((x 3Y(x - -
1)
-
zix 3) -
x(x 1) x(x) (always have
im [ 3 simplify
2x - -
=
-
- = -
to to see
+
(x 3)2(x 1)
- -
2(*3)(x 1) -
6x2 + X -
1
1) lim X+ 2 2) lim
t
4x2 4X 3
X- 8 X+
1
- -
tanx
-
L= lim 10) + 2 =
2
this means we have =lim (3x -
1)(x
+ 1)
&
X 90 O to check one-sided limits Xe -
1 (2x -
3)()
↓
= lim 3x -
1 =
3) 2) -
1
=
+
lim X+ 2 xe
= - D - 2x -
3 2( -
2) -
3
xeo-tanx >the graph of tanx tells
lim
us that tanx = 0-
-
X 90
X + 2 +
R = lim < lim tanx = o = d
Yeot tanx X - Ot
.
im Xt2 does not exist
Limits : Horizontal Asymptotes (H .
A) .
~ H A . . occur when the limit as X approaches infinity or
negative infinity g
The line y=L is called a H A
. . of the curve y=f(x) if :
lim f(x) L
=
, or lim f(x) =
L
X- * X- -
D
A
graphically
< it would look like A
-
,
:
---------------
S >
u
S 7
re
v
a
when X a proaches W W
+
& , there are no
one-sided limits lim f(x) = L lim f(x) = 1
X- -
D Xe
<
these limits will evaluate to 8 or - (indeterminant)
① When we obtain , we would divide all values in numerator & denominato
give us a value (c)
= if the highest power is in the numerator , the limit is I &
< if the
highest power is in the denominator , the limit is O
② if we have
o-o , we usually :
Mark: We will use a lot of
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