To solve the limit

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

we need to consider the behavior of the function as $x$ approaches 0 from both the left and right.

Step 1: Rewrite $\sin |x|$

Since $|x|$ is always non-negative, we can observe that $\sin |x|$ will behave similarly to $\sin x$ around $x = 0$. As $x \to 0$, $|x|$ also approaches 0.

Step 2: Apply Limit Properties

We can split the limit into two one-sided limits to analyze them separately:

1. As $x \to 0^+$: $\lim_{{x \to 0^+}} \frac{\sin |x|}{x} = \lim_{{x \to 0^+}} \frac{\sin x}{x} = 1.$

2. As $x \to 0^-$: $\lim_{{x \to 0^-}} \frac{\sin |x|}{x} = \lim_{{x \to 0^-}} \frac{\sin (-x)}{x} = \lim_{{x \to 0^-}} \frac{-\sin x}{x} = -1.$

Step 3: Conclusion

The left-hand limit and the right-hand limit do not match:

• $\lim_{{x \to 0^+}} \frac{\sin |x|}{x} = 1$,
• $\lim_{{x \to 0^-}} \frac{\sin |x|}{x} = -1$.

Since the two one-sided limits are not equal, the limit

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

does not exist.

Would you like a more detailed breakdown of any specific part of this solution?

Here are some related questions to explore:

1. What is the significance of absolute values in limit problems?
2. How does the squeeze theorem apply to trigonometric limits?
3. Can the limit exist if only one-sided limits are defined?
4. How would this limit change if we removed the absolute value?
5. How does the behavior of sine functions impact limits near zero?

Tip: When dealing with limits that involve absolute values, split the limit into one-sided cases to see if the behavior changes across the point of interest.