Question

What is the limit of the following equation?
Limit as x approaches infinity of ((e^x) - ln(x)) / (x^3)

Answer 1

Since both numerator and denominator approach infinity as x→∞, we can apply l'Hôpital's rule:
(e^x-(1/x))/3x²
Rewrite as a sum:
e^x/3x² -(1/x)/3x²
the second term goes to zero as x→∞.
Apply the rule again to the first term:
e^x/6x;
Apply the rule one last time:
e^x/6; which goes to ∞ as x→&infin.
So the limit is ∞.