lim _(xarrow infty )(x^5)/(e^x)

To evaluate the limit , we can use L'Hôpital's Rule, which is applicable for limits of the form .First, let's confirm that we have an form: Since this is indeed an form, we can apply L'Hôpital's Rule, which states that: provided the limit on the right-hand side exists.Let \(f(x) = x^5\) and \(g(x) = e^x\). We need to find the derivatives of \(f(x)\) and \(g(x)\): Now, apply L'Hôpital's Rule: We still have an form, so we apply L'Hôpital's Rule again: Again, we have an form, so we apply L'Hôpital's Rule once more: And once more: And one more time: Finally, we have: Thus, the limit is: