Q. 10 - Calculus

Question

Write the product rule for limits in terms of delta–epsilon statements. 

Step-by-Step Solution

Answer

Product rule for limits in terms of delta–epsilon statements $\left|f\right(x\left)g\right(x)-LM|<\epsilon$

Where $L=\underset{x\to c}{\lim }f\left(x\right) and M=\underset{x\to c}{\lim }g\left(x\right)$

Step 1. Given information

Let given limits $f\left(x\right) and g\left(x\right)$

Step 2. Write the product rule for limits in terms of delta&ndash;epsilon statements.

The product rule for limit is 

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

The strategy is to write the limit expression in delta-epsilon statement

For all $\epsilon >0$,there exists $\delta >0$such that if $0<|x-c|<\delta$

Then,

$$\begin{gathered} \left|\right(f\left(x\right)g\left(x\right))-(LM\left)\right|<\epsilon \\ \left|f\right(x\left)g\right(x)-LM|<\epsilon \\ Where L=\underset{x\to c}{\lim }f\left(x\right) and M=\underset{x\to c}{\lim }g\left(x\right) \end{gathered}$$