Chapter 13.2, Problem 1E

Suppose the following limits exist:

$\underset{x\to a}{\lim }f(x)\text{and}\underset{x\to a}{\lim }g(x)$

Then $\underset{x\to a}{\lim }[f(x)+g(x)]=\_\_\_\_\_\_+\_\_\_\_\_\_$, and $\underset{x\to a}{\lim }[f(x)g(x)]=\_\_\_\_\_\cdot \_\_\_\_\_$.

These formulas can be stated verbally as follows: The limit of a sum is the ______ of the limits, and the limit of a product is the _______ of the limits.

To determine

To fill: The conditions of formula for limit of a sum and limit of a product.

Answer to Problem 1E

The formula for addition and product of limits is, $\underset{x\to a}{\lim }\left[f\left(x\right)+g\left(x\right)\right]=\underset{\_}{\underset{x\to a}{\lim }f\left(x\right)}+\underset{\_}{\underset{x\to a}{\lim }g\left(x\right)}$ and $\underset{x\to a}{\lim }\left[f\left(x\right)g\left(x\right)\right]=\underset{\_}{\underset{x\to a}{\lim }f\left(x\right)\cdot \underset{x\to a}{\lim }g\left(x\right)}$, the limit of sum is the sum of the limits, and the limit of a product is the product of the limits.

Explanation of Solution

If the following limits exist, the limits are $\underset{x\to a}{\lim }f\left(x\right)$ and $\underset{x\to a}{\lim }g\left(x\right)$.

According to limit laws, if limits exist then the limit of sum of these limits is sum of the limits because both sides have same result, and limit of a product is the product of these limits because both sides have same result.

In the given statement of limit of the value of Right-Hand Side is equal to the value of Left-Hand Side for both addition and product of limits.

Thus, the formula for addition and product of limits is, $\underset{x\to a}{\lim }\left[f\left(x\right)+g\left(x\right)\right]=\underset{\_}{\underset{x\to a}{\lim }f\left(x\right)}+\underset{\_}{\underset{x\to a}{\lim }g\left(x\right)}$ and $\underset{x\to a}{\lim }\left[f\left(x\right)g\left(x\right)\right]=\underset{\_}{\underset{x\to a}{\lim }f\left(x\right)\cdot \underset{x\to a}{\lim }g\left(x\right)}$, the limit of sum is the sum of the limits, and the limit of a product is the product of the limits.