Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.2

Solution:

### Part a

By Theorem 9.3, we have

$$

\lim(x_n+y_n)=\lim x_n+\lim y_n=3+7=10.

$$

### Part b

By Theorems 9.2 and 9.3 we have

$$

\lim(3y_n-x_n)=3\lim y_n-\lim x_n=3 \cdot 7-3=18.

$$ By Theorem 9.4 we have

$$

\lim y_n^2=\left(\lim y_n\right)^2=7^2=49.

$$ It follows from Theorem 9.6 that

$$

\lim \frac{3y_n-x_n}{y_n^2}=\frac{\lim(3y_n-x_n)}{\lim y_n^2}=\frac{18}{49}.

$$