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2.8 The Exponential Limits

If then an 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)

 

Index

2.1 Modulus
2.2 Inequalities
2.3 Limits Of Functions
2.4 Left Hand And Right Hand Limits
2.5 Theorems On The Algebra Of Limits
2.6 Evaluating Limits
2.7 Limits Of Trigonometric Functions
2.8 The Exponential Limits

Chapter 3





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