Q. 9 - Calculus

Question

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

Step-by-Step Solution

Answer

The difference rule for limit is 

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

Step 1. Given information.

We need to write the difference rule $\underset{x\to c}{\lim }\left[f\right(x)-g(x\left)\right]=\underset{x\to c}{\lim }f\left(x\right)-\underset{x\to c}{\lim }g\left(x\right)$ in terms of delta-epsilon statement.

Step 2. Proof.

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

Then,

$$\begin{gathered} \left|\left[f\right(x)-g(x\left)\right]-(L-M)\right|<\epsilon \\ \left|f\left(x\right)-g\left(x\right)-L+M\right|<\epsilon \end{gathered}$$

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

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