AP Calculus AB Exam Question:

Use L'Hôpital's Rule to find the limit of the function as x approaches infinity:

Step-by-step Solution:

To apply L'Hôpital's Rule, let's first take the derivatives of the numerator and denominator:

Taking the derivatives:

Now, let's calculate the limit of the ratio of the derivatives:

Since we still have an indeterminate form of $\frac{\infty}{\infty}$, we can once again apply L'Hôpital's Rule. Taking the derivatives again:

Now, let's calculate the limit of the ratio of these new derivatives:

Since we don't have an indeterminate form anymore, we can directly evaluate the limit as x approaches infinity:

Therefore, [ \lim_{{x \to \infty}} \frac{{2e^x + x^3}}{{3e^x + x^2}} = 1 ]

Answer: The limit of the given function as x approaches infinity is equal to 1.