Final answer:
The given limit ∞/t^5 as t approaches 0 is solved by applying L'Hôpital's Rule five times, yielding a result of 1/120.
Step-by-step explanation:
To evaluate the limit lim (t->0) (e^t - 1)/t^5, we need to recognize that straightforward substitution will not work as it leads to an indeterminate form 0/0.
To solve this problem, we can apply L'Hôpital's Rule, which states that if the limit is of the form 0/0 or ∞/∞, we can take derivatives of the numerator and the denominator until we obtain a determinate form. Here, we would need to apply L'Hôpital's Rule five times since the denominator is to the fifth power.
After applying L'Hôpital's Rule five times, we calculate the fifth derivative of the numerator e^t, which is e^t again, and the fifth derivative of the denominator t^5, which is 5! (factorial of 5). Substituting t=0 into the result gives us the limit value of e^0 / 5!, which simplifies to 1/120.