Chapter 3, Problem 1RE

To determine

Whether the statement “The limit of a product is the product of the limits when each of the limits exists” is true or false.

Answer to Problem 1RE

Yes. The statement is true.

Explanation of Solution

Rule used:

Product rule:

$\underset{x\to a}{\lim }\left[f\left(x\right).g\left(x\right)\right]=\left[\underset{x\to a}{\lim }f\left(x\right)\right].\left[\underset{x\to a}{\lim }g\left(x\right)\right]=A\cdot B$ (The limit of a product is the product of the limits.)

Calculation:

Consider the function $f\left(x\right)=3x$ and $g\left(x\right)=2x^2$.

Calculate the product of the functions $f\left(x\right)$ and $g\left(x\right)$.

$\begin{array}{c}\left[f\left(x\right)\cdot g\left(x\right)\right]=3x\cdot 2x^2\\ =6x^3\end{array}$

Take limit on both sides,

$\begin{array}{c}\underset{x\to a}{\lim }\left[f\left(x\right)\cdot g\left(x\right)\right]=\underset{x\to a}{\lim }6x^3\\ =6a^3\end{array}$

Thus, $\underset{x\to a}{\lim }\left[f\left(x\right)\cdot g\left(x\right)\right]$ $=6a^3$ (1)

Take limit for the functions $f\left(x\right)=3x$ and $g\left(x\right)=2x^2$.

That is, $\underset{x\to a}{\lim }f\left(x\right)=3a$ and $\underset{x\to a}{\lim }g\left(x\right)=2a^2$.

Product the functions $f\left(x\right)$ and $g\left(x\right)$.

$\begin{array}{c}\left[\underset{x\to a}{\lim }f\left(x\right)\right]\cdot \left[\underset{x\to a}{\lim }g\left(x\right)\right]=3a\cdot 2a^2\\ =6a^3\end{array}$

Thus, $\left[\underset{x\to a}{\lim }f\left(x\right)\right]\cdot \left[\underset{x\to a}{\lim }g\left(x\right)\right]$ $=6a^3$ (2)

From the equations (1) and (2), observe that $\underset{x\to a}{\lim }\left[f\left(x\right).g\left(x\right)\right]=\left[\underset{x\to a}{\lim }f\left(x\right)\right].\left[\underset{x\to a}{\lim }g\left(x\right)\right]$.

Thus, the statement is true.