Chapter 1, Problem 2RWDT

To determine

To Conclude: Whether $\underset{x\to 0}{\lim }f\left(x\right)=0^{\prime }$ when computing some values of $f$ near $x=0$

Answer to Problem 2RWDT

No, It cannot be concluded that $\underset{x\to 0}{\lim }f\left(x\right)=0$ .

Explanation of Solution

Given:

Computing some values of $f$ near $x=0$ .

  $f\left(0.1\right)=0.1,f\left(0.01\right)=0.01,f\left(0.001\right)=0.001,f\left(0.0001\right)=0.0001\text{,}$ and

  $f\left(0.00001\right)=0.00001$ .

  $\underset{x\to 0}{\lim }f\left(x\right)=0$ ?

It is clear that the value of $x$ approaches near $0$ , the value of the function also approaches $0$ .

But all values of $x$ are greater than $0$ hence we can conclude that the right limit at

  $x=0$ .

  $\underset{x\to 0^{+}}{\lim }f\left(x\right)=0$

It is impossible to say anything about the left limit at $x=0$ .

Hence, we cannot conclude that $\underset{x\to 0}{\lim }f\left(x\right)=0$ .