1. Limits and Continuity

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (cos²x − cos x) / x²

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.

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

h. ${\lim }_{x\to 3}f\left(x\right)$

Limits of Rational Functions

In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.

f(x) = (2x + 3)/(5x + 7)

Given the function $f\(\left\)(x\(\right\))=-16x^2+64x$, complete the following. <IMAGE>

Make a conjecture about the value of the limit of the slopes of the secant lines that pass through $\(\left\)(x,f\(\left\)(x\(\right\))\(\right\))$ and $\(\left\)(2,f\(\left\)(2\(\right\))\(\right\))$ as $x$ approaches $2$.

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

g(θ)=tan πθ/10

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v'(t) = g - bv, where v(t) is the velocity of the object for t ≥ 0, g = 9.8 m/s² is the acceleration due to gravity, and b > 0 is a constant that involves the mass of the object and the air resistance.

c. Using the graph in part (b), estimate the terminal velocity lim(t→∞) v(t).