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

Consider the function

y = (1 + x)^1/x.

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.

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

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, 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.

Let = = 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:

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

The is therefore seen to lie between the two values 1, and the inequality (2) can hold only if so, 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) exists;

(3) = 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.

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 + ··· + a_n,

are continuous for every value of x.

EXERCISE 1—3

Evaluate each of the following limits: