Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 12 (c)/Solution 1

We consider the fraction of income paid as tax: $$\frac{T(x)}{x}.$$

Compute the limit as $x\to \mathrm{\infty }$: $$\underset{x\to \mathrm{\infty }}{\lim }\frac{T(x)}{x}=\underset{x\to \mathrm{\infty }}{\lim }\frac{x^3}{x(1+2x^2)}=\underset{x\to \mathrm{\infty }}{\lim }\frac{x^2}{1+2x^2}.$$

Divide numerator and denominator by $x^2$: $$\underset{x\to \mathrm{\infty }}{\lim }\frac{1}{\frac{1}{x^2}+2}=\frac{1}{2}.$$

Thus, for very large incomes, approximately $\frac{1}{2}$ of income is paid as tax.