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 (x2)  < 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
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.
