Science:Math Exam Resources/Courses/MATH102/December 2011/Question 04 (i)/Solution 1

When considering limits at infinity, only that term with the highest degree plays a role. So we can conclude instantly that the limit at either positive or negative infinity will be positive infinity.

To illustrate why this is true, consider the full computation below. We have that

${\begin{aligned}\lim _{x\to \infty }f(x)&=\lim _{x\to \infty }{\frac {1}{4}}x^{4}-2x^{3}+{\frac {9}{2}}x^{2}\\&=\lim _{x\to \infty }{\frac {1}{4}}x^{4}(1-2{\frac {1}{x}}+{\frac {9}{2x^{2}}})\\&=\lim _{x\to \infty }{\frac {1}{4}}x^{4}\\&=+\infty \end{aligned}}$