Exploring some functions

Exploring \( n^{\frac{1}{1+\delta}} \) vs \( \frac{n}{\log n} \).

n <- seq(1:1e+05)
deltas <- c(exp(1)/10, 0.001)
d <- deltas[1]
plot(n, n/log(n), type = "l", main = paste0("Delta = ", d))
lines(n, n^(1/(1 + d)), col = "blue")
legend("bottomright", legend = c("n/log(n)", "n^(1/(1+delta))"), col = c("black", 
    "blue"), bty = "n", lwd = 1)

Exploring \( n \) vs \( n^(1 + \delta) \) vs \( n log(n) \).

for (d in deltas) {
    plot(n, n, type = "l", main = paste0("Delta = ", d))
    lines(n, n^(1 + d), col = "orange")
    lines(n, n * log(n), col = "blue")
    legend("bottomright", legend = c("n", "n^(1+delta)", "n*log(n)"), col = c("black", 
        "orange", "blue"), bty = "n", lwd = 1)
}