(e ^ n times n!) / N ^ n how to find the limit x tending to infinity

(e ^ n times n!) / N ^ n how to find the limit x tending to infinity

When n tends to infinity, this limit does not exist. (e ^ n multiplied by n!) / (n ^ n multiplied by n ^ 1 / 2) the limit of n tends to infinity is (2pi) ^ 1 / 2. If the proof is long, it is not necessary

The calculation problem of the limit of x ^ 2 + 1 / x + 2 * sin1 / X tending to infinity

The limit of x ^ 2 + 1 / x + 2 * sin1 / X tends to infinity
=lim(x->∞)(x^2+1)/x(x+2)
=lim(x->∞)(1+1/x^2)/(1+2/x)
=1

Let y = f (x) be continuous on [a, positive infinity] and f (x) exist when x tends to be positive infinity. It is proved that f is bounded on [a, positive infinity]

It is proved that when x tends to positive infinity, f (x) exists, so there exists B, b > A. when x > b, | f (x) | M1
And y = f (x) is continuous on [a, positive infinity], of course it is continuous on [a, b], so when x is in the interval [a, b], | f (x) | m2
So: | f (x) | Max {M1, M2} = M