
3-46 Calculus – Differentiation and Integration
lim lim
.
( )
/
/
( )
/
/
x
x
x
x
x
x
e
e
e
e
→ + → +
−
+
=
−
+
=
−
+
=
0 0
1
1
0 0
1
1
1
1
1
1
1
1
1 0
1 0
1
As x approaches 0 through negative values, we have
1 0
1
/
/
x e
x
→ −∞ ⇒ →
lim .
( )
/
/
x
x
x
e
e
→ −
−
+
=
−
+
=−
0 0
1
1
1
1
0 1
0 1
1
Hence, we see that
lim ( ) lim ( ).
( ) ( )x x
f x f x
→ + → −
≠
0 0 0 0
So lim ( )
x
f x
®0
does not exist. Therefore, f has essential discontinuity.
Example 3.47 Verify if f is continuous at x = 0 where f is defined as f(x) = 1/ (1 - e
1/x
) when x ¹ 0
and f(0) = 0.
Solution: Now,
As x ® (0 + 0) or (0 - 0), we have 1/x ® ¥ or -¥.
Thus, as x ® (0 + 0) or (0 - 0), we have e
1/x
® ¥ or 0.
So we have
lim ( ) lim ,
lim
( ) ( )
/
(
x x
x
x