The limit provided is:

$\lim_{{x \to 0}} \frac{x}{|x|}$

Solution:

We need to analyze the behavior of this expression as $x \to 0$, which requires us to break it into two cases based on the definition of the absolute value function $|x|$.

1. When $x > 0$:

   • In this case, $|x| = x$. $\frac{x}{|x|} = \frac{x}{x} = 1$ Therefore, as $x \to 0^+$ (from the right), the limit is: $\lim_{{x \to 0^+}} \frac{x}{|x|} = 1$

2. When $x < 0$:

   • In this case, $|x| = -x$. $\frac{x}{|x|} = \frac{x}{-x} = -1$ Therefore, as $x \to 0^-$ (from the left), the limit is: $\lim_{{x \to 0^-}} \frac{x}{|x|} = -1$

Conclusion:

The left-hand limit as $x \to 0^-$ is $-1$, and the right-hand limit as $x \to 0^+$ is $1$. Since these two limits do not match, the overall limit does not exist.

So: $\lim_{{x \to 0}} \frac{x}{|x|} \text{ does not exist}.$

Would you like further details or have any questions?

Related Questions:

1. What is the formal definition of a limit not existing at a point?
2. How does the absolute value function impact limits in general?
3. What happens if the function inside the limit was $\frac{|x|}{x}$?
4. Can the limit of piecewise functions exist if one-sided limits differ?
5. How do you compute limits approaching infinity involving absolute values?

Tip:

When working with absolute value functions, always consider breaking the analysis into cases where $x$ is positive and where $x$ is negative.