I can help you with your question about expressing a limit as a definite integral.
To express a limit as a definite integral, we need to identify the function, the interval, and the subdivision of the interval. We can use the following formula as a guide:
$$\\int_a^b f(x) dx = \\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i) \\Delta x$$
where $a$ and $b$ are the lower and upper limits of the integral, $f(x)$ is the function, $x_i$ is the $i$-th point in the subdivision, and $\Delta x$ is the width of each subinterval.
In your limit, the function is $f(x) = 4x^4$, the interval is $[0, 1]$, and the subdivision is $n$ equal subintervals of width $\Delta x = 1/n$. The $i$-th point in the subdivision is $x_i = i/n$. Therefore, we can write the limit as:
$$\\lim_{n \\to \\infty} \\sum_{i=1}^n \\frac{4i^4}{n^5} = \\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i) \\Delta x$$
Using the formula above, we can express this limit as a definite integral:
$$\\int_0^1 4x^4 dx$$
I hope this helps you understand how to express a limit as a definite integral. If you have any more questions, feel free to ask me.