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Say that we are asked to prove, using the definition of limits, that the sequence ##\frac{4n^2+3}{n^2+n+2}## tends to ##4## as ##n## tends to infinity. The following is a screenshot of the solution I found in a YouTube video:
(Note that in the definition above, "g" denotes the limit  in this case ##4##).
I understand (or at least I think I do) what is happening up to the point I've marked in red  basically it appears to me that we are trying to answer the question: "for which ##n## will the distance between ##a_n## and ##4## be smaller than ##\epsilon##? This is why we are solving the absolute value inequality.
So we simplify the absolute value to ##\frac{4n+5}{n^2+n+2}##. However, I do not understand why we are saying that this is smaller than ##\frac{4n+5n}{n^2}## or why we chose ##\frac{4n+5n}{n^2}## specifically. What's the purpose behind this? Is it to try and have only one ##n## in the fraction?
For example, if we were attempting to show that ##\lim_{n \to \infty} \frac{1}{n} = 0##, we would start from ##\frac{1}{n}  0 < \epsilon## meaning that ##\frac{1}{n} < \epsilon## meaning that ##n > \frac{1}{\epsilon}##. This means that you give me ##\epsilon > 0##, and I'll give you a number ##n_0 = \frac{1}{\epsilon}##, and for all ##n > n_0## you will find that ##a_n## is within a distance of less than ##\epsilon## from ##0##, which concludes the proof (please correct me if I'm wrong here).
My question is, why can't we just do the same in the example from the screenshot above? What's the purpose of the additional step of saying that the fraction is smaller than <some intermediate value>, and why choose ##\frac{4n+5n}{n^2}## specifically as that value?
Thanks.
(Note that in the definition above, "g" denotes the limit  in this case ##4##).
I understand (or at least I think I do) what is happening up to the point I've marked in red  basically it appears to me that we are trying to answer the question: "for which ##n## will the distance between ##a_n## and ##4## be smaller than ##\epsilon##? This is why we are solving the absolute value inequality.
So we simplify the absolute value to ##\frac{4n+5}{n^2+n+2}##. However, I do not understand why we are saying that this is smaller than ##\frac{4n+5n}{n^2}## or why we chose ##\frac{4n+5n}{n^2}## specifically. What's the purpose behind this? Is it to try and have only one ##n## in the fraction?
For example, if we were attempting to show that ##\lim_{n \to \infty} \frac{1}{n} = 0##, we would start from ##\frac{1}{n}  0 < \epsilon## meaning that ##\frac{1}{n} < \epsilon## meaning that ##n > \frac{1}{\epsilon}##. This means that you give me ##\epsilon > 0##, and I'll give you a number ##n_0 = \frac{1}{\epsilon}##, and for all ##n > n_0## you will find that ##a_n## is within a distance of less than ##\epsilon## from ##0##, which concludes the proof (please correct me if I'm wrong here).
My question is, why can't we just do the same in the example from the screenshot above? What's the purpose of the additional step of saying that the fraction is smaller than <some intermediate value>, and why choose ##\frac{4n+5n}{n^2}## specifically as that value?
Thanks.
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