Function Composition Epsilon Delta

[Theocracy of the Sky]

I've been trying to solve this question for a little over an hour and am not really sure if what I'm doing is right or not. Furthermore, I'm not sure how to continue from where I've gotten

The question was "given [MATH]\lim_{x\to5} f(x)=4[/MATH] show that [MATH]\lim_{x\to5}\left[f(x)^2 - 5f(x) + 10\right] = 6[/MATH] using only epsilon-delta proofs. You are not allowed to use limit arithmetic rules".

I've attached the work I've done so far. Any and all help would be appreciated.

Thanks.

 

[Steven G]

The third line after My Answer says that | f(x) - 4| < delta. Why do you think it is true?

I would build up the results from scratch.

We know that | f(x) - 4 | < e whenever | x - 5 | < d

So -e + 4 < f(x) < e + 4 whenever | x - 5 | < d
So 5 (-e + 4 ) < 5f(x) < 5(e+4) whenever | x - 5 | < d
25(4 - e)^2 < [f(x)]^2 < 25(e+4)^2 whenever | x - 5 | < d (under what condition is this true?)
and 10 < 10 < 10 whenever | x - 5 | < d (actually 10 < 10 < 10 is always true)
Just add and see where you can go from here.