(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

f(x) = (x^{3}-11x^{2}+43x-60)/(x-4)

Prove directly from the definition that lim_{x->4}f(x)=3.

2. Relevant equations

This requires an epsilon-delta proof, I think... (will refer to epsilon as E and delta as D)

3. The attempt at a solution

Firstly I simplified the numerator of f(x) to (x-4)(x^{2}-7x+15), which enabled me to cancel the denominator.

After simplifying; f(x) = (x-4)(x-3)+3.

I want to prove that for all E>0, there exists a D>0.

Hence if |x-4|<D, then |x-4|.|x-3|+3<E.

I here assume that D≤1

Hence |x-4|≤1.

Now; |x-3| = |x-4+1|

|x-3|≤|x-4|+1

Hence |x-3|<2.

Finally,

|x-4|.|x-3|<2.|x-4|<2D

and |x-4|.|x-3|<E,

which means we choose D=min{1,E/2} i.e. delta is 1 unless E/2 is less than 1, in which case it is equal to E/2.

I want to know if i've done this correctly, and where i've gone wrong if not, also let me know if there are certain statements i should be making or ones i'm making incorrectly.

Thanks

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# Proof using Definition of Limit

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