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Example 9   Find cot x

Solution : As x® 0 the function cot x has an infinite limit. i.e. cot x = ± ¥.

To stress the fact that the function cot x can assume as x®0, both  positive  values  ( for  x >  0 ) and  negative  values  ( for x < 0 ), we write cot x = ± ¥.

Example 10  Find

Solution : The limit of the divisor i.e. (x - 2) = 0 and the limit of the dividend, i.e. (x + 4) = 2 + 4 = 6. Taking the notation in the indicated meaning, we obtain

Example 11  Find   

Solution :  

Rewriting  , the numerator tends to -1 and  the  denominator  tends  to  0 through positive values of x ( x > 0 ) from the right. Thus the function decreases without bound and

The function has a vertical asymptote at x = 0.


Example 12  Evaluate   

Solution :   In  such  examples  we  use   the  result  

Also,  for finding limit of a ratio of two polynomials in  x,  as x® ¥, we divide all terms of the ratio by the highest degree of the two polynomials. A similar procedure is also possible in many cases for fractions containing irrational terms.

\  Let    L   =  

=   

...... Dividing by x3 we get

Taking limits as x ® ¥, i.e.

L   =  

 

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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