Honors Precalculus Notes--Chapter 11, Introduction to Calculus

, Tables and Graphs
11.1
Finding Limits Using
Limit Notation and its Description




Finding Limits
Using Tables


To find limflx ) by table evaluate fat chosen values for
✗ →a
using a
,
we a .




Approach a from the left
, choosing values of ✗ that are close to a but still less than a .




Approach a from the right choosing
,
values of × that are close but still
greater than a .




Example

Find lim 4×2 We will table of values values of close 3
✗ → 3
make a
, choosing ✗ to .




xapproches3fromthelef.tt
xapproches3fromtheright.fi
> <


2.99 2.999 2.9999 3.0001 3.001 3.01


flx ) -4×2
-


35.7604 35.9760 35.9976 36.0024 36.0240 36.2404


ur




36


HI 4×2=36
>




Finding Limits Using Graphs
The limit statement limflx ) =L
✗ → a
is illustrated in the figure .




In (a) , as In (b) , as ✗ In (c) , as x




✗
approchesa, approaches a
, approchesa , fcx )

flx ) approaches flx ) approches L .

approches L . Notice,


L Ata, the This true
.
is
although however, that the


value of the f is not defined at value of the function

function is a
,
shown by the ata, f(a) is not



L : f( a) =L hole in the graph .


equal to the limit .

, Example

Use the
graph to find each of the
following :




* asx
approches -2

Iim f( × ) =
5 yapproches 5
✗→ -2



^


* Not 5 because theres a hole .


t

fl 2) -
=
3 3 because it fills in the hole .




Example
if 3

{
✗ =/
2×-4

g( × , =3
=

Graph the function -


5 if ✗




Use the
graph to find lim flx )
✗ → 3




limflx ) = 2
^
✗→ 3




f. (3) = -5

V




One -
Sided Limits