Chapter 2.2, Problem 1E

To determine

To express: the given limit statement in the form of definition and alternate definition.

Answer to Problem 1E

The definition of given limit is for every $\epsilon >0$ , there is a $\delta >0$ such that if $0<|t-c|<\delta$

, then $0<|v(t)-k|<\epsilon$

The alternate definition is for every $\epsilon >0$ , there is a $\delta >0$ such that if t is open in the

Interval $(c-\delta ,c+\delta )$ and $t\ne c$ , then $v(t)$ is in the open interval $(k-\epsilon ,k+\epsilon )$

Explanation of Solution

Given information:

The limit $\underset{t\to c}{\lim }v(t)=k$

Consider the limit $\underset{t\to c}{\lim }v(t)=k$

The above equation can be expressed as for every $\epsilon >0$ , there is a $\delta >0$ such that if

  $0<|t-c|<\delta$ , then $0<|v(t)-k|<\epsilon$

The alternate definition of given statement is

for every $\epsilon >0$ , there is a $\delta >0$ such that if t is open in the interval $(c-\delta ,c+\delta )$ and

  $t\ne c$ , then $v(t)$ is in the open interval $(k-\epsilon ,k+\epsilon )$