Real Analysis : Subsets and Limits
Let f and g be functions defined on a domain A subset or equal to R, and assume lim_x-->c f(x)=L and lim_x-->c g(x)=M for some limit point c of A then,
1-lim_x-->c k f(x)=kL for all k belong to R.
2-lim_x--> [f(x)+g(x)]=L+M
3-lim_x-->c [f(x)g(x)]=LM
4-lim_x-->c f(x)/g(x)=L/M provided M not = 0
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Let f and g be functions defined on a domain A subset or equal to R, and assume lim_x-->c f(x)=L and lim_x-->c g(x)=M for some limit point c of A then,
1-lim_x-->c k f(x)=kL for all k belong to R.
Proof. k=0 is true trivially. We assume that k is not zero. Since , for any , there exists a so that
If , then
So, we have
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Solution Summary
Proofs involving subsets and limits are provided. The solution is detailed and well presented.