limit of (x^n+nx^(n-1)+1)/e^[x]

[kinu]

\(\displaystyle \lim_{x\rightarrow \infty} \frac{x^n+nx^{n-1}+1}{e^{\left[x\right]}}\)

Where \(\left[x\right] =\) Greatest Integer Function and \(n\in\mathbb{N}\)

[João P. Ferreira]

I don't understand this notation...

What is the greatest Integer Function ??

Give me more details

[João P. Ferreira]

I've seen it

http://www.icoachmath.com/math_dictionary/Greatest_Integer_Function.html

I'm thinking...

[João P. Ferreira]

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

[kinu]

Thanks Professor

Sir can we take \(\lim_{x\rightarrow \infty}\left[x\right]\approx \lim_{x\rightarrow \infty} x\)

[josesousa]

Not in the case of limits of functions of x and [x].