In Exercises 4148, determine the one-sided limits numerically or graphically. If

What is the limit of f (x) = 1 as x ?

What is the limit of g(t) = t as t ?

Is lim x10 20 equal to 10 or 20?

Can f (x) approach a limit as x c if f (c) is undefined? If so, give an example.

What does the following table suggest about lim x1 f (x) and lim x1+ f (x)? x 0.9 0.99 0.999 1.1 1.01 1.001 f (x) 7 25 4317 3.0126 3.0047 3.00011

Can you tell whether limx5 f (x) exists from a plot of f for x > 5? Explain.

If you know in advance that limx5 f (x) exists, can you determine its value from a plot of f for all x > 5?

In Exercises 78, evaluate the limit. lim x4.2 3

In Exercises 916, verify each limit using the limit definition. For example, in Exercise 9, show that |3x 12| can be made as small as desired by taking x close to 4. lim x4 3x = 12

In Exercises 916, verify each limit using the limit definition. For example, in Exercise 9, show that |3x 12| can be made as small as desired by taking x close to 4. lim x5 3 = 3

In Exercises 916, verify each limit using the limit definition. For example, in Exercise 9, show that |3x 12| can be made as small as desired by taking x close to 4. lim x3 (5x + 2) = 17

In Exercises 916, verify each limit using the limit definition. For example, in Exercise 9, show that |3x 12| can be made as small as desired by taking x close to 4. lim x2 (7x 4) = 10

In Exercises 916, verify each limit using the limit definition. For example, in Exercise 9, show that |3x 12| can be made as small as desired by taking x close to 4. lim x0 x2 = 0

In Exercises 916, verify each limit using the limit definition. For example, in Exercise 9, show that |3x 12| can be made as small as desired by taking x close to 4. lim x0 (3x2 9) = 9

In Exercises 916, verify each limit using the limit definition. For example, in Exercise 9, show that |3x 12| can be made as small as desired by taking x close to 4. lim x0 (4x2 + 2x + 5) = 5

In Exercises 916, verify each limit using the limit definition. For example, in Exercise 9, show that |3x 12| can be made as small as desired by taking x close to 4. lim x0 (x3 + 12) = 12

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x1 x 1 x 1

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x4 2x2 32 x + 4

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x2 x2 + x 6 x2 x 2

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . im x3 x3 2x2 9 x2 2x 3

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x0 sin 2x x

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x0 sin 5x x

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim 0 cos 1

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x0 sin x x

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x4 1 (x 4)3

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x1 3 x x 1

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x3 x + 3 x2 + x 6

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x2 x + 1 x + 2

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x3+ x 4 x2 9

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim h0 3h 1 h

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim h0 sin h cos 1 h

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim h0 cos 1 h

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x0 |x| x

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x1+ sec1 x x 1

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim te t e ln t 1

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim r0 (1 + r)1/r

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x1 tan1 x cos1 x

In Exercises 1738, estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are or . lim x0 tan1 x x sin1 x x

The greatest integer function, also known as the floor function, is defined by x = n, where n is the unique integer such that n x

Determine the one-sided limits at c = 1, 2, and 4 of the function g shown in Figure 13, and state whether the limit exists at these points. 12345 1 2 3 x

In Exercises 4148, determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits areor , and describe the corresponding vertical asymptote. In Exercise 48, f (x) = x is the greatest integer function defined in Exercise 39. lim x0 sin x |x|

In Exercises 41–48, determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ? or -?, and describe the corresponding vertical asymptote. In Exercise 48, f (x) = [x] is the greatest integer function defined in Exercise 39.

In Exercises 4148, determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits areor , and describe the corresponding vertical asymptote. In Exercise 48, f (x) = x is the greatest integer function defined in Exercise 39. lim x0 x sin |x| x3

In Exercises 4148, determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits areor , and describe the corresponding vertical asymptote. In Exercise 48, f (x) = x is the greatest integer function defined in Exercise 39. lim x4 x + 1 x 4

In Exercises 4148, determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits areor , and describe the corresponding vertical asymptote. In Exercise 48, f (x) = x is the greatest integer function defined in Exercise 39. lim x2 4x2 + 7 x3 + 8

In Exercises 4148, determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits areor , and describe the corresponding vertical asymptote. In Exercise 48, f (x) = x is the greatest integer function defined in Exercise 39. lim x3 x2 x2 9

In Exercises 4148, determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits areor , and describe the corresponding vertical asymptote. In Exercise 48, f (x) = x is the greatest integer function defined in Exercise 39. lim x1 x5 + x 2 x2 + x 2

In Exercises 4148, determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits areor , and describe the corresponding vertical asymptote. In Exercise 48, f (x) = x is the greatest integer function defined in Exercise 39. lim x2 cos  2 (x x )

Determine the one-sided limits at c = 2 and c = 4 of the function f in Figure 14. What are the vertical asymptotes of f ?

Determine the infinite one- and two-sided limits in Figure 15. 5 2 4 15 5 10 x y FIGURE 14 x y 1 3 5 FIGURE 15

In Exercises 5154, sketch the graph of a function with the given limits. lim x1 f (x) = 2, lim x3 f (x) = 0, lim x3+ f (x) = 4

In Exercises 5154, sketch the graph of a function with the given limits. lim x1 f (x) = , lim x3 f (x) = 0, lim x3+ f (x) =

In Exercises 5154, sketch the graph of a function with the given limits. lim x2+ f (x) = f (2) = 3, lim x2 f (x) = 1, lim x4 f (x) = 2 = f (4)

In Exercises 5154, sketch the graph of a function with the given limits. lim x1+ f (x) = , lim x1 f (x) = 3, limx4 f (x) =

Determine the one-sided limits of the function f in Figure 16, at the points c = 1, 3, 5, 6. 1 2 3 4 1 2 3 4 5 y x 12345678 FIGURE 16 Graph of f .

Does either of the two oscillating functions in Figure 17 appear to approach a limit as x 0? (A) (B) x

In Exercises 5762, plot the function and use the graph to estimate the value of the limit. lim 0 sin 5 sin 2

In Exercises 5762, plot the function and use the graph to estimate the value of the limit. lim x0 12x 1 4x 1

In Exercises 5762, plot the function and use the graph to estimate the value of the limit. lim x0 2x cos x x

In Exercises 5762, plot the function and use the graph to estimate the value of the limit. lim 0 sin2 4 cos 1

In Exercises 5762, plot the function and use the graph to estimate the value of the limit. lim 0 cos 7 cos 5 2

In Exercises 5762, plot the function and use the graph to estimate the value of the limit. lim 0 sin2 2 sin 4 4

Let n be a positive integer. For which n are the two infinite onesided limits lim x0 1/xn equal?

Let L(n) = lim x1 n 1 xn 1 1 x for n a positive integer. Investigate L(n) numerically for several values of n, and then guess the value of L(n) in general.

In some cases, numerical investigations can be misleading. Plot f (x) = cos x . (a) Does limx0 f (x) exist? (b) Show, by evaluating f (x) at x = 1 2 , 1 4 , 1 6 ,... , that you might be able to trick your friends into believing that the limit exists and is equal to L = 1. (c) Which sequence of evaluations might trick them into believing that the limit is L = 1.

Light waves of frequency passing through a slit of width a produce aFraunhoferdiffraction pattern of light and dark fringes (Figure 18). The intensity as a function of the angle is I ( ) = Im sin(R sin ) R sin 2 where R = a/ and Im is a constant. Show that the intensity function is not defined at = 0. Then choose any two values for R and check numerically that I ( ) approaches Im as 0.

Investigate lim0 sin n numerically for several positive integer values of n. Then guess the value in general. a Intensity pattern Viewing screen Slit Incident light waves FIGURE 18 Fraunhofer diffraction pattern.

Show numerically that limx0 bx 1 x for b = 3 and b = 5 appears to equal ln 3 and ln 5, respectively, where ln x is the natural logarithm. Then make a conjecture (guess) for the value in general and test your conjecture for two additional values of b.

Investigate limx1 xn 1 xm 1 for (m, n) equal to (2, 1), (1, 2), (2, 3), and (3, 2). Then guess the value of the limit in general and check your guess for two additional pairs

Find by numerical experimentation the positive integers k such that lim x0 sin(sin2 x) xk exists.

Plot the graph of f (x) = 2x 8 x 3 . (a) Zoom in on the graph to estimate L = lim x3 f (x). (b) Explain why f (2.99999) L f (3.00001) Use this to determine L to three decimal places.

The function f (x) = 21/x 21/x 21/x + 21/x is defined for x = 0. (a) Investigate lim x0+ f (x) and lim x0 f (x) numerically. (b) Plot the graph of f and describe its behavior near x = 0.