The following conclusions can be drawn from the table.
(a)ln 1 xvaries slowly compared to x. In fact it varies more slowly than any power of x:
as x 1 → 1 0, ln 1 x 1 → 1 −∞ but x
aln 1 x 1 → 10 (3.43)
as x 1 → 1 ∞, ln 1 x 1 → 1 ∞ but x
−aln 1 x 1 → 10 (3.44)
for any positive value of a, however small.
(b)e
xvaries rapidly compared with x. In fact it varies more rapidly than any power of x:
as x 1 → 1 ∞, e
x11 → 1 ∞andx
−ae
x11 → 1 ∞ (3.45)
as x 1 → 1 ∞, e
−x11 → 1 0andx
ae
−x11 → 10 (3.46)
for any positive value of a, however large.
EXAMPLE 3.29Plots ofxe
−xandx
− 1e
xFigure 3.20 shows that whereas e
−xdecreases monotonically with increasing
value of x, the functionxe
−xfirst increases and passes through a maximum before the
exponential decay takes over. This is a characteristic behaviour of atomic orbitals. A
1 sorbital has the form e
−r, where ris the distance from the nucleus, but all other
orbitals behave with distance liker
le
−r, where lis a positive integer.
0 Exercises 43
3.9 Hyperbolic functions
Hyperbolic functions have their origin in geometry in the description of the properties
of the hyperbola. The properties of the functions are readily derived from the properties
of the other transcendental functions described in this chapter, and only a brief
discussion is presented here.
Figure 3.20
3.9 Hyperbolic functions 87
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Figure 3.21
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