Q. 24 - Calculus

Question

Limits of combinations: Fill in the blanks to complete the limit

rules that follow. You may assume that k and c are any real

numbers and that both $\underset{x\to c}{\lim }f\left(x\right) \mathrm{and} \underset{\mathrm{x}\to \mathrm{c}}{\lim }g\left(\mathrm{x}\right)$ exist.

$\underset{x\to c}{\lim }\left(f\left(x\right)g\left(x\right)\right)=\_\_\_\_\_\_$

Step-by-Step Solution

Answer

The value of the limit is $\underset{x\to c}{\lim }\left(f\left(x\right)g\left(x\right)\right)=\left(\underset{x\to c}{\lim }f\left(x\right)\right)\left(\underset{x\to c}{\lim }g\left(x\right)\right).$

Step 1. Given Information.

The given limit function is $\underset{x\to c}{\lim }\left(f\left(x\right)g\left(x\right)\right).$

Step 2. Filling the blank.

To fill the blank, use the Product Rule of limits.

So,

$\underset{x\to c}{\lim }\left(f\left(x\right)g\left(x\right)\right)=\left(\underset{x\to c}{\lim }f\left(x\right)\right)\left(\underset{x\to c}{\lim }g\left(x\right)\right).$