Solution to the Limit Problem:

We are tasked to evaluate:
$\lim_{x \to 0^+} x \cdot \cot(3x)$

1. Rewrite the cotangent function: $\cot(3x) = \frac{\cos(3x)}{\sin(3x)}$ Hence: $x \cdot \cot(3x) = x \cdot \frac{\cos(3x)}{\sin(3x)} = \frac{x \cdot \cos(3x)}{\sin(3x)}.$

2. Behavior as $x \to 0^+$:

   • As $x \to 0^+$, $\sin(3x) \to 0$, so the denominator vanishes, leading us to suspect the limit depends on the nature of the numerator and denominator's rates of vanishing.

3. Apply asymptotics (small-angle approximation): For small angles, $\sin(3x) \approx 3x$. Therefore: $\frac{x \cdot \cos(3x)}{\sin(3x)} \approx \frac{x \cdot \cos(3x)}{3x} = \frac{\cos(3x)}{3}.$

4. Evaluate the limit of $\cos(3x)$: As $x \to 0^+$, $\cos(3x) \to \cos(0) = 1$. Thus: $\lim_{x \to 0^+} x \cdot \cot(3x) = \frac{1}{3}.$

Difference Between $0^+$, $0^-$, and $0$ in $\cot x$:

1. Definition of $0^+$: $0^+$ means approaching $0$ from the right (positive) side. For $\cot x = \frac{\cos x}{\sin x}$, as $x \to 0^+$, $\sin x > 0$ and tends to $0$ positively, making $\cot x \to +\infty$.

2. Definition of $0^-$: $0^-$ means approaching $0$ from the left (negative) side. As $x \to 0^-$, $\sin x < 0$ and tends to $0$ negatively, making $\cot x \to -\infty$.

3. At $0$: The function $\cot x$ is undefined at $x = 0$ because $\sin(0) = 0$, which causes division by zero. Thus, it has a vertical asymptote at $x = 0$.

Summary:

• $0^+$: $\cot x \to +\infty$
• $0^-$: $\cot x \to -\infty$
• $0$: $\cot x$ is undefined.

Let me know if you'd like a deeper explanation or further examples!

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Related Questions:

1. What is the difference between left-hand and right-hand limits in other trigonometric functions like $\tan x$?
2. How can small-angle approximations be used for evaluating limits in trigonometric expressions?
3. How does the behavior of $\cot x$ near zero compare to $\tan x$?
4. What are other methods to evaluate $\lim_{x \to 0} f(x)$ when involving undefined trigonometric terms?
5. How does L'Hôpital's Rule apply to limits involving trigonometric functions?

Tip:

For trigonometric limits, always consider using approximations ($\sin x \approx x$, $\cos x \approx 1$) and expansions when $x \to 0$.