Show mathematically that lim t → ∞  A(t) exists if p > 1 but does not exist if 0 < p < 1.

Can you explain why lim x → ∞  A(t) exists for one of the functions but not the other?

Suppose now that p > q and a1 < p − q. This implies that an < p − q for all n (you do not needto show that, but it can be done for example by using induction, or in other ways).Find the limit of limn→∞ an. What does this mean for the population?Hint: Show that {an}∞n=1 is increasing and use a known theorem together with your result from part (a).

Select the second function, y = 1x2, for 0 < x ≤ 5.(a) Observe the graph of A(t) = t1f(x) dx in the second diagram. Does it appear that lim x → ∞ A(t) exists?YesNo    (b) Change the maximum value of x to 100 and observe the graph of A(t) again. Does it appear that lim x → ∞  A(t) exists?YesNo    If so, estimate the value of the limit. (If an answer does not exist, enter DNE.)lim x → ∞  A(t) = (c) Find the values of A(1000), A(10,000), and A(100,000).A(1000)  =  A(10,000)  =  A(100,000)  =  (d) The graphs of the first two functions (from the pull-down menu) look very similar. Can you explain why lim x → ∞  A(t) exists for one of the functions but not the other?

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)lim t → 1 1 − t1 − t

limx→∞exx