Question

Lim x → 3 (x^3 − 3x + 8) = 26
ε = 0.2 δ =
ε = 0.1 δ =

I have no idea how to do this sort of problem. Can someone show me how to solve for at least the first delta. I really need to learn it! You would SERIOUSLY be a life safe. plus i'll give a medal! :)

Answer 1

\[\|(x^3 - 3x+8)-26|<\varepsilon\\|x^3-3x-18|<\varepsilon\\|(x-3)(x^2+3x+6)|<\varepsilon\]

Answer 2

suppose \[0<|x-3|<\delta\leq 1\\-1<x-3<1\\2<x<4\\2^2<x^2<4^2\text{ and }3(2)<3x<3(4)\\4<x^2<16\text{ and } 6<3x<12\\4+6<x^2+3x<16+12\\10+6<x^2+3x+6<28+6\\16<x^2+3x+6<34\\|x^2+3x+6|<34\]

Answer 3

now\[|x-3|<\delta,\quad \|x^2+3x+6|<34,\qquad \delta \leq 1\\|x-3|\cdot|x^2+3x+6|<\delta(34),\qquad \delta\leq 1\]
put \[34\delta = \varepsilon\\\delta = \text{ minimum}\left(\frac{\varepsilon}{34},1\right)\]

note... if \(\varepsilon>34\), choose \(\delta = 1\) otherwise choose \(\displaystyle \delta=\frac{\varepsilon}{34}\)

Answer 4

just sub your \(\varepsilon\) value into the equation, and you have your answers.

Answer 5

Hello,
Thanks for taking the time to reply to my question!

The answer for this problem ( if epsilon is 0.2) is 0.0083, however I don't know how the answer came to this.

0.1 / 34 isn't the same number?

Answer 6

the answer differs from the assumed initial value of \(\delta\), which in my work, i chose \(\delta \leq 1\)

if your teacher has suggested a starting value for \(\delta\) then you should arrive at the expected result.

Answer 7

I understand. The work just says to find the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1

Answer 8

So you found a distance of 1 between x and the target point. How would we know what the maximum would be?

Answer 9

the maximum would be \(\displaystyle \frac{\varepsilon}{34}\)

Answer 10

that's not the minimum? I'm still confused how the work produced 0.0083 and yet it didn't show a starting point? :(

Answer 11

there is no minimum. the solution is given by \[\delta\leq\frac{\varepsilon}{34},\qquad \color{red}{\delta \leq 1}\]

the inequality in red produced the 34 in the denominator. a different delta, say \(\color{blue}{\delta\leq 0.1}\) will yield denominator different from 34.

Answer 12

You've been a great deal of help sir! I've learned a lot here. However, I don't know why the homework would be so vague, just poor direction? :)

But as you suggested, using delta < 0.1 gave 0.008

which is close.