Theorems on limits 3.17. If lim f(x) exists, prove that it must be unique. We must show that if lim f(x) = l, and lim f(x) = l½, then l = 12. x→x, By hypothesis, given any e >0 we can find 8 > 0 such that |f(x) – 14 | < e/2 0< |x- xol < ô when \fx) – 14|

Theorems on limits 3.17. If lim f(x) exists, prove that it must be unique. We must show that if lim f(x) = l, and lim f(x) = l½, then l = 12. x→x, By hypothesis, given any e >0 we can find 8 > 0 such that |f(x) – 14 | < e/2 0< |x- xol < ô when \fx) – 14|

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

3.17) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.

Theorems on limits
3.17.
If lim f(x) exists, prove that it must be unique.
We must show that if lim f(x) = l, and lim f(x) = l½, then l = l,.
x→x,
By hypothesis, given any e >0 we can find 8 > 0 such that
|f(x) – 14 | < e/2
0< ]x- xo] < 8
when
\f(x) – 1½] <e
< €/2
0 < |x-xol < 8
when
CHAPTER 3 Functions, Limits, and Continuity
59
Then by the absolute value property 2 on Page 4,
14 –14| = |4 -f(x) +f(x) – lz| < I4-f(x)|+ \f(x) – 1½l< e/2 + •
€/2 = €
i.e., |4-2| is less than any positive number e (however small) and so must be zero. Thus, l = l½.

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