# Real Analysis : Subsets and Limits

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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.

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

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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

...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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