Here is the correct format of the equation in the question.
A subpopulation of plant, isolated from the main population, is found to obey the function below, describing the number of individuals (in thousands).
[tex]N_{(T)} = \frac{8e^{4T}-2T+5}{7+2e^{4T}}[/tex]
What is the ultimate fate of this subpopulation of plants? Justify your claim with the appropriate mathematics.
Answer:
the ultimate fate of this subpopulation of plants = 4
Explanation:
Given that:
[tex]N_{(T)} = \frac{8e^{4T}-2T+5}{7+2e^{4T}}[/tex]
Taking the limit of N(T) ; we have ,[tex]\lim_{T \to \infty} N(T)[/tex]
[tex]N(T) = \frac{8-\frac{2T}{e^{4T}}+\frac{5}{e^{4T}}}{2+ \frac{7}{e^{4t}}}[/tex]
where T is less than [tex]e^{4T}[/tex] ; which is written as :
[tex]T< e^{4T}[/tex]
∴  [tex]\lim_{T \to \infty} N(T )= \frac{8-2 \lim_{T \to \infty} \frac{T}{e^{4T}} +5 \lim_{T \to \infty} \frac{1}{e^{4T}} }{2+7 \lim_{T \to \infty} \frac{1}{e^{4t}} }[/tex]
= [tex]\frac{8}{2}[ \lim_{T \to \infty} \frac{T}{e^{4T}} =0 ; \lim_{T \to \infty} \frac{1}{e^{4t}} }=0][/tex]
where; [tex][ \lim_{T \to \infty} \frac{T}{e^{4T}} =0 \ \ \ and \ \ \lim_{T \to \infty} \frac{1}{e^{4t}} } = 0][/tex]
Then, [tex]= \frac{8}{2}[/tex]
= 4
Thus, the ultimate fate of this subpopulation of plants = 4
