Chapter 5, Problem 4RWDT

To determine

If it is possible to have functions $f$ and $g$ such that the given conditions are satisfied.

Answer to Problem 4RWDT

It has been determined that it is not possible to have functions $f$ and $g$ such that $\underset{x\to c}{\lim }f\left(x\right)$ and $\underset{x\to c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)$ both exist but $\underset{x\to c}{\lim }g\left(x\right)$ does not exist.

Explanation of Solution

Given:

  $\underset{x\to c}{\lim }f\left(x\right)$ and $\underset{x\to c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)$ both exist but $\underset{x\to c}{\lim }g\left(x\right)$ does not exist.

Concept used:

According to the property of limits,

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

Calculation:

Let $\underset{x\to c}{\lim }f\left(x\right)$ and $\underset{x\to c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)$ both exist.

Then, let $\underset{x\to c}{\lim }f\left(x\right)=p$ and let $\underset{x\to c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)=r$ , where $p$ and $r$ are finite real values.

Put these values in $\underset{x\to c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)=\underset{x\to c}{\lim }f\left(x\right)-\underset{x\to c}{\lim }g\left(x\right)$ to get,

  $r=p-\underset{x\to c}{\lim }g\left(x\right)$

Simplifying,

  $\underset{x\to c}{\lim }g\left(x\right)=p-r$

Since $p$ and $r$ are finite real values, it follows that $p-r$ is also a finite real value.

This implies that $\underset{x\to c}{\lim }g\left(x\right)$ exists.

Hence, if $\underset{x\to c}{\lim }f\left(x\right)$ and $\underset{x\to c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)$ both exist, then $\underset{x\to c}{\lim }g\left(x\right)$ also exists.

So, it is not possible to have functions $f$ and $g$ such that $\underset{x\to c}{\lim }f\left(x\right)$ and $\underset{x\to c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)$ both exist but $\underset{x\to c}{\lim }g\left(x\right)$ does not exist.

Conclusion:

It has been determined that it is not possible to have functions $f$ and $g$ such that $\underset{x\to c}{\lim }f\left(x\right)$ and $\underset{x\to c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)$ both exist but $\underset{x\to c}{\lim }g\left(x\right)$ does not exist.