When Central Limit Theorem (CLT) Fails?

library(tidyverse)
library(Rcpp)
library(patchwork)

n = 30
num_samples = 1000

Normal dist:
normal_means = rnorm(num_samples, mean = 0, sd = 1 / sqrt(n))

cauchy Dist:
sourceCpp('cauchy_dist.cpp')

cauchy_means = dist_cauchy(n = n, num_samples = num_samples)

mydata = tibble(
 dist_norm = normal_means,
 dist_cauchy = cauchy_means
)

p1 =  
 ggplot(data =   mydata, aes(x = dist_norm)) +
 geom_density(fill = 'orange') +
 labs(title = 'Normal Distribution', x = NULL, y = NULL) +
 theme_minimal()

p2 =
 ggplot(data =   mydata, aes(x = dist_cauchy)) +
 geom_density(fill = 'orange') +
 labs(title = 'Cauchy Distribution', x = NULL, y = NULL) +
 theme_minimal()

p1 + p2