Chapter A3, Problem 20E

To determine

To show: The given condition.

Answer to Problem 20E

Here, by the principal of mathematical induction, it is true for all $\text{n}$ .

Explanation of Solution

Given: The function $f_1\left(x\right),f_2\left(x\right)$ and $f_3\left(x\right)$ has limits $L_1,L_2$ and $L_3$ as $x\to c$ then $\underset{x\to c}{\lim }\left[f_1\left(x\right)\cdot f_2\left(x\right)\cdot \dots f_n\left(x\right)\right]=L_1\cdot L_2\cdot \dots L_n$ .

Calculation:

Use the principle of mathematical induction.

First prove the case for $n=1$ .

Step 1: For $n=1,\underset{x\to c}{\lim }{f}_1\left(x\right)=L_1$ as given.

Step 2: Suppose $\underset{x\to c}{\lim }\left[f_1\left(x\right)\cdot f_2\left(x\right)\cdot \dots f_k\left(x\right)\right]=L_1\cdot L_2\cdot \dots L_k$

  $\begin{array}{l}\underset{x\to c}{\lim }\left[f_1\left(x\right)\cdot f_2\left(x\right)\cdot \dots f_{k+1}\left(x\right)\right]=\underset{x\to c}{\lim }\left[f_1\left(x\right)\cdot f_2\left(x\right)\cdot \dots f_k\left(x\right)\right]\cdot L_{k+1}\\ \text{ =}{L}_1\cdot L_2\cdot \dots L_{k+1}\end{array}$

Here, the product rule is used.

Hence, by principal of mathematical induction, this is true for all $\text{n}$ .

Conclusion:

Hence, by principal of mathematical induction, this is true for all $\text{n}$ .