Projected learning outcomes: Attempt to interpret and understand the definition of a limit This will give the main idea behind the mathematical proof of the limit of in Equation 1, as the continuity of functions of the form f(x) = frequently, further illustrating that there are will use in this course as well 0. We will refer back to this fact proof) behind the "facts" we directly) mx + 6 for m (i.e we will not have time to cover every such reason logical reasons а (even though Problems 1. Adapt the work on the first example from Supplemental Lecture 2 to show by the definition of the limit that for any real numbers a, b, m with m / 0, we have that lim (mx (1) та + b. х—а Hint: First state what you wish to accomplish by interpretting the meaning of Equation (1). Then use your interpretation and the guidance from the supplemental lecture notes to find positive function 5(e) which works. Recall that, when multiplying inequality by a number, you keep track of whether this number is positive or negative (since such an operation will preserve or reverse the inequality, respectively) а dividing both sides of or an Lucture Supplament 2 We ux he rigorous detiaitim of a lmit a few ean.pls Exnafla' Ue definim of imtto shou thatli 2x = 2. x1 We st find 04 /x-i1<5Ce) guarau tes 2 x -2

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 8E

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Example number 1 on the second image is where 'Example' appears first followed by the example.

Projected learning outcomes: Attempt to interpret and understand the definition of a limit
This will give the main idea behind the mathematical proof of the limit of in Equation 1,
as the continuity of functions of the form f(x) =
frequently, further illustrating that there are
will use in this course
as well
0. We will refer back to this fact
proof) behind the "facts" we
directly)
mx + 6 for m
(i.e
we will not have time to cover every such reason
logical
reasons
а
(even though
Problems
1. Adapt the work on the first example from Supplemental Lecture 2 to show by the
definition of the limit that for any real numbers a, b, m with m / 0, we have that
lim (mx
(1)
та + b.
х—а
Hint: First state what you wish to accomplish by interpretting the meaning of Equation (1).
Then use your interpretation and the guidance from the supplemental lecture notes to find
positive function 5(e) which works. Recall that, when multiplying
inequality by a number, you keep track of whether this number is positive or negative (since
such an operation will preserve or reverse the inequality, respectively)
а
dividing both sides of
or
an

Lucture Supplament 2
We ux
he rigorous detiaitim of a lmit
a few ean.pls
Exnafla' Ue definim of
imtto
shou thatli 2x = 2.
x1
We st find
04 /x-i1<5Ce) guarau tes
2 x -2<E
positive fnetion E
that
Phnt
Frmunty to bid a quess r5
we worle acewods
First
ne consider that
2x-21=2(x-) = 121-I-1 2-l< 1
w 2x-1,
Sioe 20
ued to
Since 270
we may divida
2 while
pradring
both sides of this
e inquality
JK-1
take 5)=
Thus
we
0 k-1<
what that
tt fallows tt when
2-21=12(a-)= 12l- 2«1x-|522 =E.
Sine 20
Tolt
As a
ol
e hawe
Pamare Fov asns
5- prots are preated in fhe opporife ordkr
Haveer, at thic leel
pertminivg to mutlernatical logic,
mere Concerred
we
Cfe
fmhion Ce widh weras
Plene beadrare at there is are addifiona dresks
to be made to rewrite this as a
proof

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