Question

Let g(x) = x2 – x – 7 be a function from the real numbers to the real numbers. Find the set of δ values that satisfy the formal definition of limx4 g(x) = 5 when given the value ε = 0.3

Answer 1

i would say it largely depends on what \(g(x)\) is

Answer 2

Can you please show me how to solve this showing each step so I can complete my other homework questions

Answer 3

ok now we have a question

Answer 4

\[\lim_{x\to 4}g(x)=5\] means exactly that given any \(\epsilon>0\) there is a \(\delta\) (which you write in terms of \(\epsilon\) ) such that if \(|x-4|<\delta\) then \(|g(x)-5|<\epsilon\)

Answer 5

you were not given a general \(\epsilon\) you were given \(\epsilon=0.3\) so your job is to find \(\delta\) if \(|x-4|<\delta\) then
\[|x^2-x-7-5|<.03\]

Answer 6

this should work out more or less nicely since no doubt when you compute the thing inside the absolute values signs you will be able to factor it, and one of the factors will be \(x-4\) which is what you have control over

Answer 7

in fact you can just about see it already
\[|x^2-x-7-5|=|x^2-x-12|=|(x-4)(x+3)|=|x-4||x+3|\]

Answer 8

how are we doing so far? almost done

Answer 9

I'm still confused but I will keep rereading until I understand it fully. Thank you for your help. Much appreciated!

Answer 10

first off you should know what exactly you have to do
then go from the general \(|g(x)-L|<\epsilon\) to the specific \(|x^2-x-7-5|<0.3\)

Answer 11

then it is some algebra to see what you need
once you get to \(|x-4||x+3|<.3\) figure out how small \(|x-4|\) must be in order to make this true

Answer 12

you have control over the size of \(|x-4|\) in particular you can say \(|x-4|<1\) making
\[-1<x-4<1\iff 3<x<5\] making \(|x+3|<8\)

Answer 13

then let \(|x-4|<\frac{.3}{8}\) and you will have what you need

Answer 14

Thank you

Answer 15

you can probably even do better, but as you get to pick \(\delta\) no need to be generous, make it nice and small