Limit & Natural log of determinant.

Question

Question:

my attempt:

i expand determinant along first column

$\lim_{x\to\infty}\left[x.\ln\left(\dfrac{a}{x^3}+b-\dfrac{c}{x}\right)\right]=-5$

$\lim_{t\to0}\left[\dfrac{\ln\left({at^3}+b-{ct}\right)}{t}\right]=-5$

after this i don't know how to proceed further

i'm trying to make argument of $\ln$ "simple" such that after expanding it we can compare coefficient of both sides and get values of $a,b,c$. But since question asks for value of $b+c$ i think there might be some method which can give directly the sum without explicitly calculating all values. thank you for your help

Answer 1

In order for $$ \lim_{t\to 0}\left[\frac{\ln(at^3+b-ct)}{t}\right]=-5, $$ the term inside the log must converge to $1$, producing a $0/0$ limit. This tells you $b=1$. For the rest, try applying L'Hospital's rule to the term on the left. You should be able to solve for $c$ directly through one equation involving $b$ and $c$, with $b$ known.