2.8 The Exponential Limits
If then a_{n}
is monotonic increasing and bounded, then
and
If we put
, then as
Hence
and
We know that 'e' is an irrational number and 2 < 3 < 3. Also, we shall assume some results without proof.
and
The term log a on R.H.S. has base 'e' which need not be explicitly written.
The students should note that the power of 'a' in the numerator viz 'x' must be repeated in the denominator and it must tend to zero. For this sometimes, the denominator has to be readjusted as follows.
.... (Since 4x
® 0 as x ® 0 )
Example 1
Find
Solution : L =
=
=
=
= 2 (log 3) (1)
=
= log 9
Example 2
Find
Solution : L =
=
=
=
= log a  log b
= log (a / b)
