# 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

Â© BrainMass Inc. brainmass.com March 6, 2023, 1:28 pm ad1c9bdddfhttps://brainmass.com/math/real-analysis/real-analysis-subsets-limits-27193

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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

...

#### Solution Summary

Proofs involving subsets and limits are provided. The solution is detailed and well presented.