Unformatted text preview: ´ x)  f (x)
or Î´ x â†’ 0
Î´x
h
We were actually taking the limit from the right.
imit
In other words, we were taking xLâ†’ a+ i.e. approaching the instant a
from the right.
We should ask: what would happen if we took L imit i.e. approaching the L imit hâ†’ 0 x â†’ a instant a from the left ?
f (x  h )  f (x)
or
 h
because x â†’ a i n terms of Î´ x is :
this is L imit hâ†’ 0 L imit Î´x â†’ 0 f (x  Î´ x)  f (x)
 Î´ x x â†’ a = a  Î´ x as Î´ x â†’ 0 and (a  Î´ x )  a =  Î´ x .
Will we get the same INSTANTANEOUS RATE OF CHANGE ?
In general, for any function f(x) we expect:
INSTANTANEOUS RATE OF CHANGE
INSTANTANEOUS RATE OF CHANGE
= approaching a from the right
approaching a from the left
If they are equal then we say that f(x) is DIFFERENTIABLE at x = a. L imit x â†’ a L imit Î´x â†’ 0 f (x)  f (a)
=
x  a f (a  Î´ x)  f (a)
=
 Î´ x 86 L imit x â†’ a+ L imit Î´x â†’ 0 f (x)  f (a)
x  a
f (a + Î´ x)  f (a)
Î´x Example 1: is f(x) differentiable at x = a ?
1
differentiable
f(x)
a f (x ) = { x for 0 â‰¤ x â‰¤ a
for
2a  x for a â‰¤ x â‰¤ 2 a a
0,0)
(0,0 )
Approaching a from the left : f(x) = x
left
Near a and to the left of a :
left 2a x x = a âˆ’ Î´x f(x) âˆ’ f(a)
L imit f (a âˆ’ Î´x) âˆ’ f(a)
= Î´x â†’ 0
xâˆ’a
(a âˆ’ Î´x) âˆ’ a
L imit (a âˆ’ Î´x) âˆ’ a
L imit âˆ’ Î´x
= Î´x â†’ 0
= Î´x â†’ 0
= +1
âˆ’ Î´x
âˆ’ Î´x Limit x â†’ aâˆ’ Approaching a from the right : f(x) = 2a âˆ’ x
right
Near a and to the right of a : x = a + Î´x
right Limit x â†’ a+ f(x) âˆ’ f(a)
L imit f (a + Î´x) âˆ’ f(a)
= Î´x â†’ 0
xâˆ’a
(a + Î´x) âˆ’ a { 2a âˆ’(a +Î´x)} âˆ’ { 2a âˆ’a }
L
L imit âˆ’ Î´x
= Î´ximit0
= Î´x â†’ 0
= âˆ’1
â†’
+ Î´x
+Î´x
f(x) âˆ’ f(a)
f(x) âˆ’ f(a)
left derivative Limit âˆ’
â‰ right derivative Limit +
right
xâ†’a
xâ†’a
xâˆ’a
xâˆ’a
we say : f(x) is not differentiable at x = a.
differentiable
87 Example 2 : is f(x) = x differentiable at x = 0 ?
differentiable
f(x) f(x) = x (0 , 0 )
Approaching 0 from the left : f(x) = x = âˆ’x
left
Near 0 and to the left...
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 Fall '09
 TAMERDOÄŸAN
 Limit, Î”x

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