Question

Calculus definition of a limit!

Let f (x)= 4x-5
(a) Estimate L, where lim x-->2 = L
(b) ε=0.2 determine the preimage (L-ε , L+ε) write your answer using interval notation.
(c) Does the preimage that you found in part (b) contain a deleted open interval centered at 2? If so, what is the maximum possible radius of the deleted open interval? Your answer should be exact, not approximate.

So I understand part a. so ignore that. But I don't understand how to do parts (b)and (c). The answer for (b)=(1.95,2.05) and (c)= yes; 0.05 Can anyone explain how this is so? Please and thank you!

Answer 1

sorry i need to read on preimage, I never did this sort of approximation with delta epsilon proofs...

Answer 2

hmmm... your answers look good... what's the part that confuses you?

Answer 3

I don't understand how to get the answers for parts (b) and (c). Our teacher gave the answers on the bottom of the worksheet, but idk how to get it and how to make the preimage

Answer 4

the so-called preimage will be the epsilon given, 0.2, added to the limit and subtracted

Answer 5

so the \( \bf lim_{x\to 2} 4x-5 = 3\)

the preimages will be, using the epsilon quantity of 0.2 up and down our limit, of 3

or, what is "x" when y = 3 + 0.2 = 3.2 and y = 3-0.2 = 2.8

Answer 6

sorta.... my graph is hehehe, not-to-scale hehe

Answer 7

Now I get it. The question is wrong. The preimage refers to the subset of domain (an interval) but part b) clearly implies that the student is being asked to find the "image" or the interval (2.8, 3.2) which is a subset of the range, when the question should actually be asking to determine the interval (x - δ, x + δ) where δ = 0.05.

@Ikushinkan

Answer 8

agree on that, is more semantics issue than anything else
their wording is not the greatest

Answer 9

"(c) Does the preimage that you found in part (b) contain a deleted open interval centered at 2? "


they really mean

so, if you use 3.2 and 2.8, you'd get an interval at \(\bf \delta\) what are those 2 open intervals? when y = 3.2 and y = 2.8, what is "x"

Answer 10

well, as your answer shows, if you plug in the values for y = 3.2, you'd get 2.05
that makes an interval of [2.05, 2)

if you plug in the values for y = 2.8, you'd get 1.95
that makes an interval of (2, 1.95]

Answer 11

what's the GAP between, 2, where "x" is moving towards, and those numbers?
well, the GAP is \(\boldsymbol{\delta}\) which, as you can see

from 1.95 to 2, well, there's only 0.05
from 2 to 2.05, well, also 0.05

which means \(\boldsymbol{\delta}= 0.05\)