Photoreceptors are cells in the retina that change their voltage in response to light. These changes in voltage are relayed to the brain via the optic nerve, and thus we see things. In response to a bright but short pulse of light (a bright flash, say), the voltage response of a photoreceptor can sometimes be accurately described by the function V(t) = t n e −kt , for some constants n and k > 0.

a. What is limt→∞ V(t)? Does this limit depend on the value of n? (Note: this isn’t easy to answer analytically, so we suggest you approach this by drawing a series of graphs and seeing how they behave.)

b. Based on your answer to the previous part of the question, what is lim t→∞
e t + 4t 4 e −t + t 3 4e t − 4t 3e −t + t 2 ?
Justify your answer carefully.