Q. 25 - 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.

$\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\_\_\_\_\_\_, \mathrm{provided} \mathrm{that} \mathrm{\_\_\_\_\_}$

Step-by-Step Solution

Answer

The value of the limit is $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{lim}}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}{\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{lim}}\mathbf{g}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}, \mathrm{provided} \mathrm{that} \underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{lim}}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{\ne }\mathbf{0}.$

Step 1. Given Information.

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

Step 2. Filling the blank.

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

So,

$\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{\mathrm{x}\to \mathrm{c}}{\lim }\mathrm{f}\left(\mathrm{x}\right)}{\underset{\mathrm{x}\to \mathrm{c}}{\lim }\mathrm{g}\left(\mathrm{x}\right)}, \mathrm{provided} \mathrm{that} \underset{\mathrm{x}\to \mathrm{c}}{\lim }g\left(x\right)\ne 0.$