I've seen it
http://www.icoachmath.com/math_dictionary/Greatest_Integer_Function.html
I'm thinking...
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| limit of (x^n+nx^(n-1)+1)/e^[x] https://forumdematematica.org/viewtopic.php?f=7&t=150 |
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| Autor: | kinu [ 26 jan 2012, 07:46 ] |
| Título da Pergunta: | limit of (x^n+nx^(n-1)+1)/e^[x] |
\(\displaystyle \lim_{x\rightarrow \infty} \frac{x^n+nx^{n-1}+1}{e^{\left[x\right]}}\) Where \(\left[x\right] =\) Greatest Integer Function and \(n\in\mathbb{N}\) |
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| Autor: | João P. Ferreira [ 27 jan 2012, 00:44 ] |
| Título da Pergunta: | Re: limit |
I don't understand this notation... What is the greatest Integer Function ?? Give me more details |
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| Autor: | João P. Ferreira [ 27 jan 2012, 00:46 ] |
| Título da Pergunta: | Re: limit |
I've seen it http://www.icoachmath.com/math_dictionary/Greatest_Integer_Function.html I'm thinking... |
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| Autor: | João P. Ferreira [ 27 jan 2012, 01:24 ] |
| Título da Pergunta: | Re: limit |
Realize that \(x-1<\left[x\right]<x+1\) \(e^{x-1}<e^{\left[x\right]}<e^{x+1}\) \(\frac{1}{e^{x-1}}>\frac{1}{e^{\left[x\right]}}>\frac{1}{e^{x+1}}\) So we can conclude that: \(\frac{x^n+nx^{n-1}+1}{e^{x-1}}>\frac{x^n+nx^{n-1}+1}{e^{\left[x\right]}}>\frac{x^n+nx^{n-1}+1}{e^{x+1}}\) \(\frac{ex^n+enx^{n-1}+e}{e^x}>\frac{x^n+nx^{n-1}+1}{e^{\left[x\right]}}>\frac{(1/e)x^n+(1/e)nx^{n-1}+1/e}{e^x}\) Now we apply the limits.. \(\lim_{x \to \infty}\frac{ex^n+enx^{n-1}+e}{e^x}\ >\ \lim_{x \to \infty} \frac{x^n+nx^{n-1}+1}{e^{\left[x\right]}}\ >\ \lim_{x \to +\infty} \frac{(1/e)x^n+(1/e)nx^{n-1}+1/e}{e^x}\) Because in the fraction we have polynomial funtions on the numerator and exponential functions on the denominator those limits are zero. \(0 > \lim_{x \to \infty} \frac{x^n+nx^{n-1}+1}{e^{\left[x\right]}} >0\) Then that limit is zero... Take care |
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| Autor: | kinu [ 27 jan 2012, 16:58 ] |
| Título da Pergunta: | Re: limit |
Thanks Professor Sir can we take \(\lim_{x\rightarrow \infty}\left[x\right]\approx \lim_{x\rightarrow \infty} x\) |
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| Autor: | josesousa [ 27 jan 2012, 20:17 ] |
| Título da Pergunta: | Re: limit |
Not in the case of limits of functions of x and [x]. |
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