The limit in the image you've provided is:

$\lim_{x \to 0^+} \frac{x^5}{5} \cdot \cos\left(\frac{3\pi}{x}\right)$

To evaluate this limit, let's break it down:

1. As $x \to 0^+$, the term $\frac{x^5}{5}$ approaches 0, since any small positive value raised to the power of 5 will approach 0.

2. The term $\cos\left(\frac{3\pi}{x}\right)$ oscillates between -1 and 1 as $x \to 0^+$, because the cosine function always stays between -1 and 1 regardless of the input.

3. The product of a term approaching 0 ($\frac{x^5}{5}$) and a bounded term ($\cos\left(\frac{3\pi}{x}\right)$) will approach 0, since the zero factor dominates.

Thus, the limit is:

$\lim_{x \to 0^+} \frac{x^5}{5} \cdot \cos\left(\frac{3\pi}{x}\right) = 0$

Would you like further clarification on this? Let me know if you have more questions.

Related Questions:

1. What happens if the exponent of $x$ is increased in this limit?
2. How would the result change if we used a sine function instead of cosine?
3. How can you handle limits of oscillatory functions like this more generally?
4. What other techniques could be applied to solve similar limit problems?
5. How can we prove rigorously that $\cos\left(\frac{3\pi}{x}\right)$ is bounded?

Tip: When evaluating limits involving oscillating functions, focus on the behavior of other parts of the expression that tend toward a specific value (like 0) as they can "cancel out" the effect of oscillations.