Evaluate the expression limx−→∞ x2ex

To evaluate the limit of the expression as x approaches infinity, we can use L'Hopital's Rule. This rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.

The expression is in the form of f(x)/g(x) where f(x) = x^2 and g(x) = e^x.

First, we need to find the derivatives of f(x) and g(x).

The derivative of f(x) = x^2 is f'(x) = 2x. The derivative of g(x) = e^x is g'(x) = e^x.

Now, we apply L'Hopital's Rule:

lim (x→∞) [f(x)/g(x)] = lim (x→∞) [f'(x)/g'(x)] = lim (x→∞) [2x/e^x]

We can see that the new limit is still in the form of 0/0 or ∞/∞, so we can apply L'Hopital's Rule again.

The derivative of f'(x) = 2x is f''(x) = 2. The derivative of g'(x) = e^x is g''(x) = e^x.

So, we have:

lim (x→∞) [f'(x)/g'(x)] = lim (x→∞) [f''(x)/g''(x)] = lim (x→∞) [2/e^x]

As x approaches infinity, e^x approaches infinity much faster than 2, so the limit of the expression is 0.