Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \lim _{x \rightarrow 0} \frac{|x|}{x}=1 $$

Short answer

The statement is false. The 2-sided limit of this function as \(x\) approaches 0 doesn't exist. From the positive side, the limit is 1, while from the negative side, it's -1.

Step-by-step solution

Understanding the function

Absolute function is represented as \(|x|\). It's the magnitude of \(x\) without regard to its sign. So, when we divide \(|x|\) by \(x\), we're essentially evaluating two separate conditions: one where \(x > 0\), and another where \(x < 0\). Therefore, the original function can be rewritten as \(\frac{|x|}{x} = \frac{x}{x}\) for \(x > 0\) and \(\frac{|x|}{x} = \frac{-x}{x}\) for \(x < 0\).

Calculating the limit from the positive direction

Now we calculate the limit from the positive direction. For \(x > 0\), the function simplifies to \(\frac{x}{x} = 1\). So the limit as \(x\) approaches 0 from the positive side (\(x \rightarrow 0^+\)) is 1. That is, \(\lim_{x\rightarrow 0^+}\frac{|x|}{x}=1\)

Calculating the limit from the negative direction

Next, we calculate the limit from the negative direction. For \(x < 0\), the function simplifies to \(\frac{-x}{x} = -1\). So the limit as \(x\) approaches 0 from the negative side (\(x \rightarrow 0^-\)) is -1. That is, \(\lim_{x\rightarrow 0^-}\frac{|x|}{x}=-1\)

Conclusion

Here it's clear that the 2-sided limit (as \(x\) approaches 0 from both, the positive and the negative side) doesn't exist since the limit from the positive side isn't equal to the limit from the negative side. So, the given statement is false.