i)When x goes to infinity we get:
[tex]\lim _{x\rightarrow+\infty}P(x)=\lim _{x\rightarrow+\infty}\frac{3x+1}{x+4}[/tex]Computing the limit we get:
[tex]\lim _{x\rightarrow+\infty}\frac{3x+1}{x+4}=\lim _{x\rightarrow+\infty}\frac{\frac{3x}{x}+\frac{1}{x}}{\frac{x}{x}+\frac{4}{x}}=\lim _{x\rightarrow+\infty}\frac{3+\frac{1}{x}}{1+\frac{4}{x}}=\frac{3+0}{1+0}=3[/tex]Therefore the fish population tends to 3,000 fishes as x goes to infinity.
ii) The initial population is:
[tex]P(0)=\frac{3\cdot0+1}{0+4}=\frac{1}{4}[/tex]which is equal to 1000/4=250 fishes.
iii) This model is restricted to x≥0 because there were no fishes of the same type in the pond before the introduction of the species.
