Let f (x) = x2 cos 1 x , x _= 0 (a) Use a graphing calculator to sketch the graph of y =

Let f (x) = x2 cos 1 x , x _= 0 (a) Use a graphing calculator to sketch the graph of y = f (x). (b) Show that x2 x2 cos 1 x x2 holds for x _= 0 . (c) Use your result in (b) and the sandwich theorem to show that lim x0 x2 cos 1 x = 0

Let f (x) = x cos 1 x , x _= 0 (a) Use a graphing calculator to sketch the graph of y = f (x). (b) Use the sandwich theorem to show that lim x0 x cos 1 x = 0

Let f (x) = ln x x , x > 0 (a) Use a graphing calculator to graph y = f (x). (b) Use a graphing calculator to investigate the values of x for which 1 x ln x x 1 x holds. (c) Use your result in (b) to explain why the following is true: lim x ln x x = 0

Let f (x) = sin x x , x > 0 (a) Use a graphing calculator to graph y = f (x). (b) Explain why you cannot use the basic rules for finding limits to compute lim x sin x x (c) Show that 1 x sin x x 1 x holds for x > 0, and use the sandwich theorem to compute lim x sin x

In Problems 520, evaluate the trigonometric limits. lim x0 sin(2x) 2x

In Problems 520, evaluate the trigonometric limits. lim x0 sin(2x) 3x

In Problems 520, evaluate the trigonometric limits. lim x0 sin(5x) x

In Problems 520, evaluate the trigonometric limits. lim x0 sin x x

In Problems 520, evaluate the trigonometric limits. lim x0 sin( x) x

In Problems 520, evaluate the trigonometric limits. lim x0 sin(x/2) 2x

In Problems 520, evaluate the trigonometric limits. lim x0 sin(x) x

In Problems 520, evaluate the trigonometric limits. lim x0 sin2 x x

In Problems 520, evaluate the trigonometric limits. lim x0 sin x cos x x(1 x)

In Problems 520, evaluate the trigonometric limits. lim x0 1 cos2 x x2

In Problems 520, evaluate the trigonometric limits. lim x0 1 cos x 2x

In Problems 520, evaluate the trigonometric limits. lim x0 1 cos(2x) 3x

In Problems 520, evaluate the trigonometric limits. lim x0 1 cos(5x) 2x

In Problems 520, evaluate the trigonometric limits. lim x0 1 cos(x/2) x

In Problems 520, evaluate the trigonometric limits. lim x0 sin x(1 cos x) x2

In Problems 520, evaluate the trigonometric limits. lim x0 csc x cot x x csc x

(a) Use a graphing calculator to sketch the graph of f (x) = eax sin x, x 0 for a = 0.1, 0.01, 0, 0.01, and 0.1. (b) Which part of the function f (x) produces the oscillations that you see in the graphs sketched in (a)? (c) Describe in words the effect that the value of a has on the shape of the graph of f (x). (d) Graph f (x) = eax sin x, g(x) = eax , and h(x) = eax together in one coordinate system for (i) a = 0.1 and (ii) a = 0.1. [Use separate coordinate systems for (i) and (ii).] Explain what you see in each case. Show that eax eax sin x eax Use this pair of inequalities to determine the values of a for which lim x f (x) exists, and find the limiting value.