Question

Lim┬(n→∞)⁡⁡〖(∜(x^5+2)-∛(x^2+1))/(√(5&x^4+2)-√(x^3+1))〗

Answer 1

Ответ:

345

Пошаговое объяснение:

Answer 2

To evaluate the limit of the expression as n approaches infinity:

Lim┬(n→∞)⁡⁡〖(∜(x^5+2)-∛(x^2+1))/(√(5&x^4+2)-√(x^3+1))〗

Let's simplify the expression step by step:

First, we'll deal with the numerator: ∜(x^5+2) = (√(x^5+2))^(1/4) = ((x^5+2)^(1/2))^(1/4) = ((x^5+2)^(1/2))^(1/4) = (x^5+2)^(1/8)

∛(x^2+1) = (x^2+1)^(1/3)

Now let's simplify the denominator: √(5x^4+2) = (√(5x^4+2)) = (√(5(x^4+2/5))) = (√(5)√(x^4+2/5)) = (√(5))((x^4+2/5)^(1/2)) = (√(5))((x^4)^(1/2))(2/5)^(1/2) = (√(5))(x^2)(√(2/5))

√(x^3+1) = (x^3+1)^(1/2)

Now let's substitute these simplified expressions back into the original expression:

Lim┬(n→∞)⁡⁡〖(∜(x^5+2)-∛(x^2+1))/(√(5&x^4+2)-√(x^3+1))〗

= Lim┬(n→∞)⁡⁡〖((x^5+2)^(1/8) - (x^2+1)^(1/3))/((√(5))(x^2)(√(2/5)) - (x^3+1)^(1/2))〗

Now, as n approaches infinity, we need to check the exponents of the highest powers of x in the numerator and denominator. We'll focus on those terms since they will dominate the expression:

For the numerator:

• The highest power of x is x^5.

For the denominator:

• The highest power of x is x^3.

Since the exponent of x is larger in the numerator, we can conclude that the limit as n approaches infinity will be ∞ or undefined.

Therefore, the limit of the given expression as n approaches infinity does not exist or is infinite (∞).