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Example 1  Find the right hand and the left hand limits of a function 'f ' as follows:

Solution : When x > 4 | x - 4 | = x - 4

Example 2  Show that 4x = 8

Solution :  For  | 4x - 8 | < e    if    | 4 (x-2) | < e

i.e.  if    4 | x - 2 | < e   i.e.,    if    | x - 2 | < e/4

Thus d = e/4 ;   we find,   therefore,   that every e> 0, a number d > 0 where d = e/4,   satisfying   | 4x - 8 | < e   for   all | x - 2 | < d

Hence 4x = 8

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Example 3  If f (x) = , x ¹ 0    f (x) DNE. Prove this statement.

Solution : If x > 0, we have f (x) = = 1

\ f (x ) = 1

Example 4   Show that sin , x ¹ 0 is oscillating between -1 and +1.

Solution :  Let sin ,    x ¹ 0

We observe that

and

Thus

Further we know that -1 £ sin  £ 1

\ The graph of y = sin lies between y = -1 and y = +1 and oscillates up and down very rapidly as x®0. This behavior of the function is expressed by saying that the limit of f(x) as x®0 DNE and it is further described by saying that f(x) is oscillatory, the limits of oscillation being -1 and +1.

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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