B if when any number δ however large is assigned we

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B. If, when any number Δ , however large, is assigned, we can choose y 0 (Δ) so that φ ( y ) > Δ when 0 < y 5 y 0 (Δ) , then we say that φ ( y ) tends to as y tends to 0 by positive values, and we write φ ( y ) → ∞ . We define in a similar way the meaning of ‘ φ ( y ) tends to the limit l as y tends to 0 by negative values’, or ‘lim φ ( y ) = l when y → - 0’. We have in fact only to alter 0 < y 5 y 0 ( ) to - y 0 ( ) 5 y < 0 in definition A. There is of course a corresponding analogue of definition B, and similar definitions in which φ ( y ) → -∞ as y +0 or y → - 0. If lim y +0 φ ( y ) = l and lim y →- 0 φ ( y ) = l , we write simply lim y 0 φ ( y ) = l.
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[V : 94] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 202 This case is so important that it is worth while to give a formal definition. If, when any positive number , however small, is assigned, we can choose y 0 ( ) so that, for all values of y different from zero but numerically less than or equal to y 0 ( ) , φ ( y ) differs from l by less than , then we say that φ ( y ) tends to the limit l as y tends to 0 , and write lim y 0 φ ( y ) = l. So also, if φ ( y ) → ∞ as y +0 and also as y → - 0, we say that φ ( y ) → ∞ as y 0. We define in a similar manner the statement that φ ( y ) → -∞ as y 0. Finally, if φ ( y ) does not tend to a limit, or to , or to -∞ , as y +0, we say that φ ( y ) oscillates as y +0, finitely or infinitely as the case may be; and we define oscillation as y → - 0 in a similar manner. The preceding definitions have been stated in terms of a variable de- noted by y : what letter is used is of course immaterial, and we may suppose x written instead of y throughout them. 94. Limits as x tends to a . Suppose that φ ( y ) l as y 0, and write y = x - a, φ ( y ) = φ ( x - a ) = ψ ( x ) . If y 0 then x a and ψ ( x ) l , and we are naturally led to write lim x a ψ ( x ) = l, or simply lim ψ ( x ) = l or ψ ( x ) l , and to say that ψ ( x ) tends to the limit l as x tends to a . The meaning of this equation may be formally and directly defined as follows: if, given , we can always determine δ ( ) so that | φ ( x ) - l | < when 0 < | x - a | 5 δ ( ) , then lim x a φ ( x ) = l.
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[V : 96] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 203 By restricting ourselves to values of x greater than a , i.e. by replacing 0 < | x - a | 5 δ ( ) by a < x 5 a + δ ( ), we define ‘ φ ( x ) tends to l when x approaches a from the right’, which we may write as lim x a +0 φ ( x ) = l. In the same way we can define the meaning of lim x a - 0 φ ( x ) = l. Thus lim x a φ ( x ) = l is equivalent to the two assertions lim x a +0 φ ( x ) = l, lim x a - 0 φ ( x ) = l. We can give similar definitions referring to the cases in which φ ( x ) → ∞ or φ ( x ) → -∞ as x a through values greater or less than a ; but it is probably unnecessary to dwell further on these definitions, since they are exactly similar to those stated above in the special case when a = 0, and we can always discuss the behaviour of φ ( x ) as x a by putting x - a = y and supposing that y 0.
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