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In a HS problem, we study the functions

$$f_n:x longmapsto (x-n)ln x - xln(x-n)$$ for $n$ natural with $nge 5$

First questions just ask the domain, the monotonicity ($f_n$ is defined for $x> n$ and is strictly decreasing on its domain) then by IVT we show that the equation $f_n(x)=0$ has a unique solution $alpha_n$ that satisifes $n+1 < alpha_n <n+2$.

From last inequality it's clear that $(alpha_n)$ diverges to $+infty$.

The last question asks to prove that $limlimits_n to +infty (alpha_n - n)=1$ and then to evaluate $limlimits_n to +infty (alpha_n+1 - alpha_n)$.

I'm stuck with this last question.

For the first part note that

$$fracalpha_n-nlog(alpha_n - n) = fracalpha_nlog(alpha_n) $$

Since $alpha_n to infty$, the right hand side diverges to infinity so the left hand side must diverge to infinity as well. Since $1 < alpha_n - n < 2$ for all $n$, this implies that $log(alpha_n-n) to 0$ i.e. $alpha_n - n to 1$.

For the second part

$$ beginalign lim(alpha_n+1 - alpha_n) & = lim((a_n+1 - (n+1)) - (alpha_n - n) + 1) \ & = lim(alpha_n+1 - (n+1)) - lim(alpha_n - n) + 1 \ & = 1 - 1 + 1 \ & = 1 endalign$$

This could be a stupid answer.

You look for the zero of
$$f_n=(x-n)log( x) - xlog(x-n)$$ and you showed that the root is between $n+1$ and $n+2$. So,let $x=n+1+epsilon$ to make
$$f_n=(1+epsilon ) log (n+1+epsilon)-(n+1+epsilon) log (1+epsilon)$$ Expand it as a Taylor series for infinitely large values of $n$ to get
$$f_n=-n log (1+epsilon)-(1+epsilon)log left(frac1+epsilon nright)+Oleft(frac1nright)$$ So, if $nto infty$, $epsilonto 0$