For if x 0 then ψ x 1 1 while if 0 x 1 or 1 x 0 then

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For if x = 0 then ψ ( x ) = [1] = 1; while if 0 < x < 1, or - 1 < x < 0, then 0 < 1 - x 2 < 1 and so ψ ( x ) = [1 - x 2 ] = 0. Or again, let us consider the function y = x/x already discussed in Ch. II , § 24 , (2). This function is equal to 1 for all values of x save x = 0. It is not equal to 1 when x = 0: it is in fact not defined at all for x = 0. For when we say that φ ( x ) is defined for x = 0 we mean (as we explained in Ch. II , l.c. ) that we can calculate its value for x = 0 by putting x = 0 in the actual expression of φ ( x ). In
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[V : 97] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 207 this case we cannot. When we put x = 0 in φ ( x ) we obtain 0 / 0, which is a meaningless expression. The reader may object ‘divide numerator and denominator by x ’. But he must admit that when x = 0 this is impossible. Thus y = x/x is a function which differs from y = 1 solely in that it is not defined for x = 0. None the less lim( x/x ) = 1 , for x/x is equal to 1 so long as x differs from zero, however small the difference may be. Similarly φ ( x ) = { ( x + 1) 2 - 1 } /x = x + 2 so long as x is not equal to zero, but is undefined when x = 0. None the less lim φ ( x ) = 2. On the other hand there is of course nothing to prevent the limit of φ ( x ) as x tends to zero from being equal to φ (0), the value of φ ( x ) for x = 0. Thus if φ ( x ) = x then φ (0) = 0 and lim φ ( x ) = 0. This is in fact, from a practical point of view, i.e. from the point of view of what most frequently occurs in applications, the ordinary case. Examples XXXVI. 1. lim x a ( x 2 - a 2 ) / ( x - a ) = 2 a . 2. lim x a ( x m - a m ) / ( x - a ) = ma m - 1 , if m is any integer (zero included). 3. Show that the result of Ex. 2 remains true for all rational values of m , provided a is positive. [This follows at once from the inequalities (9) and (10) of § 74 .] 4. lim x 1 ( x 7 - 2 x 5 + 1) / ( x 3 - 3 x 2 + 2) = 1. [Observe that x - 1 is a factor of both numerator and denominator.] 5. Discuss the behaviour of φ ( x ) = ( a 0 x m + a 1 x m +1 + · · · + a k x m + k ) / ( b 0 x n + b 1 x n +1 + · · · + b l x n + l ) as x tends to 0 by positive or negative values. [If m > n , lim φ ( x ) = 0. If m = n , lim φ ( x ) = a 0 /b 0 . If m < n and n - m is even, φ ( x ) + or φ ( x ) → -∞ according as a 0 /b 0 > 0 or a 0 /b 0 < 0. If m < n and n - m is odd, φ ( x ) + as x +0 and φ ( x ) → -∞ as x → - 0, or φ ( x ) → -∞ as x +0 and φ ( x ) + as x → - 0, according as a 0 /b 0 > 0 or a 0 /b 0 < 0.]
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[V : 97] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 208 6. Orders of smallness . When x is small x 2 is very much smaller, x 3 much smaller still, and so on: in other words lim x 0 ( x 2 /x ) = 0 , lim x 0 ( x 3 /x 2 ) = 0 , . . . . Another way of stating the matter is to say that, when x tends to 0, x 2 , x 3 , . . . all also tend to 0, but x 2 tends to 0 more rapidly than x , x 3 than x 2 , and so on. It is convenient to have some scale by which to measure the rapidity with which a function, whose limit, as x tends to 0, is 0, diminishes with x , and it is natural to take the simple functions x , x 2 , x 3 , . . . as the measures of our scale.
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