Mathematics

Problem

Consider the functions

f(x)=3+\dfrac{7}{x^{4}}

and

g(x)=\bigl(3+\dfrac{7}{x^{4}}\bigr)\bigl(-2+\dfrac{1}{x^{2}}\bigr)

. (i) Explain, without performing detailed numerical calculation, why

\displaystyle \lim_{x\to\infty}f(x)=3

. (ii) Using similar reasoning, determine

\displaystyle \lim_{x\to\infty}g(x)

and state the property of limits that justifies every step of your explanation.