Under what constraints , if any, does the binomial distribution become equal to the normal distribution? I understand that when n approaches infinity binomial distribution also approaches a Poisson distribution. What about the normal distribution? I googled and found that if n approaches infinity and p and q are large (not close to either 1 or 0) then it follows a normal distribution. But if p is large np should also be infinite. So it seems to me that normal distribution can only approximate the binomial distribution when n, p, and q all are finite. Is my reasoning correct?

Traditionally, normal approximations have been used to
to get serviceable answers to such problems as finding
$P(X \le 7),$ where $X \sim \mathsf{Binom}(n = 30, p = 1/6).$
Extensive tables of binomial CDFs are relatively rare, and using the binomial PDF formula to find the necessary eight terms
seems excessively tedious.
Several rules of thumb have been proposed to limit the use
of normal approximations to situations in which something like two place accuracy is possible. One of the most common
guidelines is that $\min[np, n(1-p)] \ge 5,$ which helps to
ensure that the approximating normal distribution puts most of its probability on $(0,n).$ It is not always mentioned that approximations tend to be best if $p$ is near $1/2,$ so that the binomial distribution
is nearly symmetrical.
The normal approximation for the specific problem above is
as follows:
$$P(X \le 7) = P(X < 7.5) =
P\left( 
\frac{X-np} { \sqrt{np(1-p)} } < 
\frac{7.5 - 5}{ \sqrt{25/6} }
= 1.225\right)\\ \approx P(Z < 1.225) = 0.8897,$$
where $Z$ is standard normal, the first step is known as a continuity correction,
and the last would require interpolation, using most
printed standard normal CDF tables.
In R, an exact computation is simply $P(X \le 7) = 0.8863.$
pbinom(7, 30, 1/6)
[1] 0.8863132

In the figure below, the exact probability is the sum of the
heights of the vertical bars to the left of the dotted
vertical line. The normal approximation is the area
under the density curve to the left of that line.

R code for figure:
x = 0:30;  pdf = dbinom(x, 30, 1/6)
hdr = "BINOM(30, 1/6) with Normal Approx"
plot(x, pdf, type="h", lwd=3, col="blue", main=hdr)
 abline(h=0, col="green2")
 abline(v=0, col="green2")
 abline(v=7.5, col="orange", lwd=2, lty="dotted")
curve(dnorm(x, 5, sqrt(25/6)), add=T, lwd=2, col="brown") 

Note: A Poisson approximation to a binomial probability is often useful when $n$ is large and the Poisson mean $\mu = np$ is of moderate size. In the current example, $n$ is not large enough for a good Poisson approximation. $P(Y \le 7) = 0.8666$, where $Y\sim\mathsf{Pois}(\lambda = 5).$
ppois(7, 5)
[1] 0.8666283
k = 0:7;  sum(exp(-5)*5^k/factorial(k))
[1] 0.8666283

