Find the limit of (x ^ (1 + x)) / ((1 + x) ^ x) - X / E at positive infinity of X,

Find the limit of (x ^ (1 + x)) / ((1 + x) ^ x) - X / E at positive infinity of X,

X ^ (1 + x) / (1 + x) ^ x = x ^ X / (1 + x) ^ x * x = x / (1 + 1 / x) ^ x original formula = x [1 / (1 + 1 / x) ^ X - 1 / E] = x [e - (1 + 1 / x) ^ x] / [e (1 + 1 / x) ^ x] = x [e - (1 + 1 / x) ^ x] / e ^ 2 = 1 / e ^ 2 * [e - (1 + T) ^ (1 / t)] / T = 1 / e ^ 2 * [e - (E - (ET) / 2 + (11et ^ 2) / 24 - O (T ^ 3

Find the limit of (x ^ n) / e ^ ax when x tends to infinity

a>0
lim (x^n)/e^ax
=lim (nx^n-1)/ae^ax
=lim (n(n-1)x^n-2)/a ² e^ax
.
=lim n!/ a^ne^ax
=0
a

Why can't the limit of (x + 6) e ^ (1 / x) - x (x tends to positive infinity) be 6? Because the limit of e ^ (1 / x) is 1, and then the original formula is x + 6-x, which is 6. What's wrong with this

x→∞lim(x+6)e^(1/x)-x
=x→∞lim{[xe^(1/x)-1]+6(e^(1/x)}
=x→∞lim[xe^(1/x)-1]+6
=x→∞6+lim{[e^(1/x)-1]/(1/x)}
0 / 0 formula
=x→∞6+lim{e^(1/x)}
=6+1
=7

When x tends to infinity, why is the limit of (1 + 1 / x) ^ x not 1, but e?

Let t = 1 / x, then s = (1 + 1 / x) ^ x = (1 + T) ^ (1 / T), X tends to ∞, then t tends to 0
LNs = ln (1 + T) / T, when t tends to 0, the numerator denominator tends to 0, so the derivation of the numerator denominator can be obtained by using the Robida rule
Then LNs tends to 1 / (1 + T) = 1, obviously s tends to E

It is proved that if the function f (x, y) is continuous on the bounded closed region D, the function g (x, y) is integrable on D, and G (x, y) ≥ 0, (x, y) belongs to D, then at least one point (a, b) belongs to D, so that ∫ ∫ (region D) f (x, y) g (x, y) d Δ= F (a, b) ∫ ∫ (area D) g (x, y) d Δ

Because f (x, y) is continuous on the bounded closed region D, f has a minimum m and a maximum m;
Then m * ∫ ∫ (area D) g (x, y) d Δ=< ∫∫ (area D) f (x, y) g (x, y) d Δ<= M * ∫ ∫ (area D) g (x, y) d Δ; Then, according to the intermediate value theorem of continuous function, at least one point (a, b) belongs to D, so that ∫ ∫ (region D) f (x, y) g (x, y) d Δ= F (a, b) ∫ ∫ (area D) g (x, y) d Δ.

When finding the proof process (n) of the limit, Lim1 / 2 ^ 2 = 0; (x → 2) LIM (x + 3) = 5; (x →∞) Lim1 / x = 0 Sorry, I'm self-taught now. I can't understand the examples,

Let: x = 1 + T (T - > 0) LIM (x - > 1) [x ^ (n + 1) - (n + 1) x + n] / (x-1) ^ 2 = LIM (T - > 0) [(1 + T) ^ (n + 1) - (n + 1) (1 + T) + n] / T ^ 2 = LIM (T - > 0) [[1 + (n + 1) t + (n + 1) n / 2T ^ 2 + O (T ^ 2)] - (n + 1) - (n + 1) t + n] / T ^ 2 = (n + 1) n / 2