Why can a term be factored out of the numerator in a converging sequence?

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SUMMARY

The sequence defined by the expression (n+1)^(1/2) / (2(n)^(1/2)) converges to 1/2. The discussion clarifies that both the numerator and denominator must be analyzed when determining convergence and limits. Specifically, the term (n)^(1/2) can be factored out from the numerator, which is essential for understanding the behavior of the sequence as n approaches infinity. This factorization is valid because it simplifies the expression and reveals the limit more clearly.

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trap101
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State whether the sequence converges and if so, find the limit


(n+1)1/2/2(n)1/2

ok so I got that it converges to 1/2, my question more so lies in the fact that why are we able to factor out a (n)1/2 from the term in the numerator? Isn't it only the denominator that we are concerned about when it comes to powers? or is it because the denominator is technically "factored" already?
 
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trap101 said:
State whether the sequence converges and if so, find the limit


(n+1)1/2/2(n)1/2

ok so I got that it converges to 1/2, my question more so lies in the fact that why are we able to factor out a (n)1/2 from the term in the numerator? Isn't it only the denominator that we are concerned about when it comes to powers?
No, you also need to look at the numerator.
trap101 said:
or is it because the denominator is technically "factored" already?
 
ok. thanks
 

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