1.1 Limits

1 . 1 limits


limits at a point 2) one-sided limits
D

1) two-sided limits lim f(x) lim f(x)
=
x= a X- at
-X + lef+
analytical ex : lim x right *
X- Z


22 2 +1 3 eX
=1
-
= :




graphical ex : look at
graph
lim
tabular ex : use calculator = DNE because

table function (plug in a)

Vertical Asymptotes limits at infining
eX : lim * = x Horizontal Asymptotes
*
X - ot lim * =
0 lim * = 0
*

D X
x X- -




#

ex : lim * =
-D shows and behavior
*
D



· X-0- If lim f(x) = the same #

VA :
X = 0 **o then the # ·


·


If one or both limits at a is IX is a HA

then a is a VA



examples
limf(x) b) limf(x) =
2 a) f(1) = 3
-


as =
1
* -3




I
X - - X+ -
2t
-
2


-
I


I 9) limf(x) = 0 as limf(x) = -
* +) limf(x) =
3
in3z
I
I

-is M
-




X * 8 X Xp -
x




· -- 2 #




9) lim f(x) = 0 b) f(0) = 1 2) limf(x) = 1

*
X 8 X- zt


& ⑧ ·
d) limf(x) = 2 e) limf(x) =* f) lim f(x) =
1

X-2- X-x X-X