Q. 26 - 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.

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

Step-by-Step Solution

Answer

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

Step 1. Given Information.

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

Step 2. Filling the blank.

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

So,

$\underset{x\to c}{\lim }f\left(g\right(x\left)\right)=f\left(\underset{x\to c}{\lim }g\left(x\right)\right), \mathrm{provided} \mathrm{that} f \mathrm{is} \mathrm{continuous} \mathrm{at} \underset{x\to c}{\lim }g\left(x\right).$