Question

Evaluate the limit, using l'hôpital's rule. enter inf for [infinity], -inf for - [infinity], or dne if the limit does not exist, but is neither [infinity] nor -[infinity]

lim 4eˣ - 4x - 4 / 13x² = ________

Answer 1

Using L'Hôpital's Rule, the limit of (4e^x - 4x - 4) / (13x^2) as x approaches infinity is found to be infinity due to the exponential growth of e^x outpacing the quadratic growth of the denominator.

Step-by-step explanation:

The student is asked to evaluate a limit and may use L'Hôpital's Rule to do so. For the function given by (4ex - 4x - 4) / (13x2), as x approaches infinity, the limit could approach infinity, a finite number, or it might not exist. When the forms 0/0 or ∞/∞ arise, L'Hôpital's Rule can be applied by taking derivatives of the numerator and the denominator separately and then evaluating the limit of the new function. Here, since the degree of the exponential function in the numerator is always higher than the polynomial in the denominator, the limit will be infinity. Therefore, applying L'Hôpital's Rule repeatedly will show that the limit is indeed infinity.

As x approaches infinity, the exponential function grows much faster than any polynomial, and thus the limit of (4ex - 4x - 4) / (13x2) as x approaches infinity is infinity.