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}.
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.
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; PT_{1} and PT_{2} are tangents at T_{1} and T_{2}. From trigonometry, we have:
T_{1}ST_{2} < T_{1}MT_{2} < T_{1}PT_{2};
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) = a_{0}x^{n} + a_{1}x^{n}^{−1} + a_{2}x^{n}^{−2} + ··· + a_{n},
are continuous for every value of x.
EXERCISE 1—3
Evaluate each of the following limits: