SOME SPECIAL LIMITS - Functions, Rates, and Limits - The Calculus Primer

The Calculus Primer (2011)

Part I. Functions, Rates, and Limits

Chapter 4. SOME SPECIAL LIMITS

1—18. The Limit of (1 + x)1/x. One of the most useful limits in the calculus is the

images

Consider the function

y = (1 + x)1/x.

x

y

10

1.001

2

1.732

1

2.000

.5

2.250

.1

2.594

.01

2.705

.001

2.717

−.5

4.000

−.1

2.868

−.01

2.732

−.001

2.720

From graphic considerations, therefore, it appears that as x → 0, the limit of y, or of (1 + x)1/x, lies between 2.717 and 2.720. By using more advanced methods (Chapter Eleven), we shall learn how to compute this limit e to any number of decimal places. Note that y approaches e as a limit not only as x approaches 0 from the right, but also as x approaches 0 from the left as well.

images

1—19. The Limit of images. Another useful limit is

images

Reference to a five-place (or more) table of values of trigonometric functions will reveal that for all angles less than 2°, the sine of the angle and the angle itself (expressed in radians) are very nearly equal. In fact, for all angles less than 10°, the angle in radians and its sine are equal to three decimal places. Thus for small angles, the sine and the angle are nearly equal; the smaller the angle, the less the difference between them becomes. Hence we see that as x→ 0, images appears to be equal to 1.

That this actually is the limit can be proved from the following geometrical proof. Let O be the center of a circle whose radius is unity.

images

Let images = images = x; PT1 and PT2 are tangents at T1 and T2. From trigonometry, we have:

T1ST2 < T1MT2 < T1PT2;

hence2 sin x < 2x < 2 tan x.(1)

Dividing (1) through by 2 sin x, we get:

images

Now as x → 0, the above inequality (2) holds true for all values of x, however small. At the same time, as x → 0,

images

The images is therefore seen to lie between the two values 1, and the inequality (2) can hold only if images so, images by the properties of inequalities and reciprocals.

1—20. Continuity of a Function in an Interval. We are now in a position to formulate what is meant by the continuity of a function in an interval more precisely than was stated in §1—6.

A single-valued function f(x) is said to be continuous at x = a if :

(1)f(a) is defined;

(2)images exists;

(3)images = f(a).

If any one of these three conditions is not satisfied, the function is said to be discontinuous at the point where x = a. If the reader should find this formal definition somewhat complicated at first, he will find that it will become more significant as he becomes more familiar with the subject.

images

In general, a function is said to be continuous throughout an interval if it is continuous at every point in the interval. More specifically, polynomial functions, such as

f(x) = a0xn + a1xn−1 + a2xn−2 + ··· + an,

are continuous for every value of x.

EXERCISE 1—3

Evaluate each of the following limits:

images

images

images