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I got stucked while I was working out whether $$\frac{4n-3}{n+43}$$ converges. I would be pleased if I could a hint of the above question.

Arturo Magidin
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Nhlakanipho
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    Could you possibly format your question so it will not refer to the title, perhaps include some of your own work and say what did you try and where did you get stuck? – Asaf Karagila Apr 29 '11 at 17:48
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    Hint: $$\frac{4n+3}{n+43}=\frac{n(4+\frac{3}{n})}{n(1+\frac{43}{n})}$$ – Apostolos Apr 29 '11 at 17:55
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    http://math.stackexchange.com/questions/33970/finding-the-limit-of-fracqnpn-where-q-p-are-polynomials/33971#33971 –  Apr 29 '11 at 17:56

2 Answers2

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Hint: $$\frac{4n+3}{n+43} = 4-\frac{175}{n+43} $$

lhf
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What are the tools that you can use? One possible way is to use Cauchy's theorem: Show that for any given $\varepsilon>0$ there exists $N>0$ such that for all $n>N$ we have $\left|\frac{4n-3}{n+43}-4\right|<\varepsilon$

Dennis Gulko
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    correct answer, but I'd just add that the term "Cauchy's theorem" is non-standard in this context; most people would just say, "by the definition of limits". – Gerben Apr 29 '11 at 20:23