What is the limit of the sum of the x power of E and the negative x power of E? Can infinity and infinitesimal be added to find the limit?

What is the limit of the sum of the x power of E and the negative x power of E? Can infinity and infinitesimal be added to find the limit?

Let y = e ^ x + e ^ (- x) = e ^ x + 1 / e ^ X
Because e ^ x > 0, when x tends to 0, the minimum value of this equation is 2,
When x tends to infinity, because e ^ x tends to infinity and e ^ - x tends to 0, the limit of the whole formula is infinity
That is to say, when infinity and infinitesimal are added to find the limit, its limit is infinity

Find the answer to a limit problem of high number!
limxe-1/x /x
x→0+
Note: - 1 / X is the superscript of E, xe-1 / X divided by X as a whole, when x → 0+
When the limit, the answer I know is 0, how do you calculate it?... waiting for expert advice··

Is that what the title means?
lim [x*e^(-1/x)]/x
x→0+
The original form
=lim e^(-1/x)=lim [1/e^(1/x)]
x→0+ x→0+
When x → 0 +, (1 / x) → + ∞
Then e ^ (1 / x) → + ∞
1/e^(1/x)→0
That is Lim [x * e ^ (- 1 / x)] / x = 0
x→0+

1. Let xn = cos (n π / 2) / N ask LIM (x →∞) xn =? To find out n, so that when n > N, the absolute value of the difference between xn and its limit is less than the positive number δ, when δ = 0.001, the number n is found out
2. Prove that (1 + A & sup2 / N & sup2;) = 1 under the root sign of LIM (x →∞)
3. If LIM (x →∞) UN = a, it is proved that LIM (x →∞) ▏ UN ▕ = ▏ a ▕, and an example is given to illustrate that when the sequence ▏ UN ▕ converges, the sequence u may not converge

1. LIM (n →∞) cos (n π / 2) / N = 1. LIM (. N →∞) xn = 0. When solving N, n must satisfy 1 / n