Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.3

Solution: By Theorem 9.4, we have

$$

\lim a_n^3=(\lim a_n)^3=a^3,\quad \lim b_n^2=(\lim b_n)^2=b^2

$$ By Theorem 9.2, we have

$$

\lim 4a_n=4\lim a_n=4a.

$$ Therefore, we have, by Theorem 9.4,

$$

\lim (a_n^3+4a_n)=\lim a_n^3+\lim 4a_n=a^3+4a

$$ and

$$

\lim (b_n^2+1)=\lim b_n^2 +1=b^2+1.

$$ Clearly, we have $\lim (b_n^2+1)>0$, hence we can use Theorem 9.6 and obtain

$$

\lim s_n=\frac{\lim(a_n^3+4a_n)}{\lim(b_n^2+1)}=\frac{a^3+4a}{b^2+1}.

$$ We are done.