[Technology Exercise] Let f(t) = 1/t for t≠0.
         
a. Find the average rate of change of f with respect to t over the intervals (i) from t=2 to t=3, and (ii) from t=2 to t=T.
         
b. Make a table of values of the average rate of change of f with respect to t over the interval [2,T], for some values of T approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001, and 2.000001.
         
c. What does your table indicate is the rate of change of f with respect to t at t=2?

Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.

a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all vertical asymptotes.

x² + x ― 6

c. y = ------------------

x² + 2x ― 8

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.

a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.

b. About how fast was the object going when it hit the surface?

[Technology Exercise] 22. Make a table of values for the function at the points x=1.2, x=11/10, x=101/100, x=1001/1000, x=10001/10000, and x = 1.

a. Find the average rate of change of F(x) over the intervals [1,x] for each x≠1 in your table.

b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1.

The accompanying graph shows the total distance s traveled by a bicyclist after t hours.

b. Estimate the bicyclist’s instantaneous speed at the times t=1/2, t=2, and t=3.

The accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after it has been driven for t days.

c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

Existence of Limits

In Exercises 5 and 6, explain why the limits do not exist.

limx→0 x/|x|

Suppose that a function f(x) is defined for all real values of x except x=c. Can anything be said about the existence of limx→c f(x)? Give reasons for your answer.