Q. 8 - Calculus

Question

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

Step-by-Step Solution

Answer

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

$\left|kf\left(x\right)-L\right|<\epsilon$

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

Step 1. Given information.

The constant multiple rule for limit is 

$\underset{x\to c}{\lim (k}f\left(x\right))=k\underset{x\to c}{\lim }f(x)$, for any real number k.

Step 2. Statement.

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

Then,

$\left|kf\left(x\right)-L\right|<\epsilon$

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