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Unformatted text preview: ∆ in the right or in the left
x
direction.
When the tangent of the angle at which the tangent line at point x0 to
When
the graph of function y = f ( x ) is intersecting the horizontal axis
depends on the sign of ∆ x , the derivative of f ( x ) at x0 is not defined.
In that case we can talk about the “lefthand side slope” and the
In
“righthand side slope” or, more formally, the leftside and rightside
leftside
derivatives.
derivatives.
Notation for the rightside derivatives: f ′( x0 + 0 ) = lim
+ f ( x0 + ∆x ) − f ( x0 )
∆x f ′( x0 − 0 ) = lim− f ( x0 + ∆x ) − f ( x0 )
∆x x → x0 The leftside derivates: x → x0 12 An Example x, x ≥ 0
Consider a function y = f ( x ) = x = − x , x < 0
This function has a kink at (0,0): Let us compute the rightside
Let
derivative at point x0 = 0 : f ′( 0 + 0 ) = lim+
x →0 (0,0) 0 + ∆x − 0
=1
∆x Now compute the leftside
Now
derivative at point x0 = 0 : f ′( 0 − 0 ) = lim+
x →0 0 − ∆x − 0
= −1
∆x Even if the derivative of x does not exist at x=0, the leftside
Even
and rightside derivatives do exist and are well defined.
13 LeftSided and RightSided Limits
Limits are not only taken for the difference quotients in order to compute
Limits
the derivatives. We can talk in general about a limit of any function of
any variable:
any lim g ( v ) = g ( v0 ) = q0 v→ 0
v Limits are answering questions of the type, “what value does variable
q=g(v) approach as variable v approaches v0 ?
q=g(v)
In some cases the answer depends on the direction in which v
In
approaches v0
The leftside limit of q is symbolized by: g ( v0 − 0) = vlim g ( v )
The
−
→ v0
The rightside limit of q: g ( v0 + 0 ) = lim g ( v )
+
v →v0 14 Graphical Illustration of OneSided Limits
q=g(v) 2, v < 5
This is a step function: q = g ( v ) = 1, v ≥ 5
As v approaches 5 from the left, the leftside limit
As
of g(v) is 2.
of
However, as v approaches 5 from the right,
However,
the rightside limit of g(v) is 1.
the 2 The limit proper of g(v) at v=5 does not
The
exist, but the two onesided (unequal)
limits do.
limits 1 5 v 15 Evaluation of the Limits
() To evaluate a limit of function f x at point x0 it suffices to make sure
To
that the right and the leftside limits at this point are equal to each
other.
This would be the case e.g. for
This y = 2 + x2 However, what do we do when
However, x0 and any real x0 is not even in the domain of f ( x ) ? Consider the following function that is not defined at x=1: 1− x2
y = f ( x) =
1− x We can still evaluate the value of the limit at x=1 even if the
We
function is not defined at x=1 (since you can’t divide by zero) by
simplifying the expression:
simplifying 1 − x 2 (1 − x ) (1 + x )
y = f ( x) =
=
= 1 + x, x ≠ 1
1− x
1− x lim f ( x ) =2
x→
1 16 Ratio of Two Infinities
Consider finding the limit of f ( x ) =
Consider It is difficult to evaluate f
It
two infinities is equal to. 2x + 5
at x0 = + ∞
x +1 ( + ∞ ) since one is not sure what the ratio of Once a...
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 Fall '11
 Kim

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