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# Permachart - Marketing Reference Guide: Inflection Point, Inflection, Farad

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LIMITS
l e a r n r e f e r e n c e r e v i e w
Differential Calculus
Differential Calculus
FUNCTIONS
EXPLICIT FUNCTIONS
L’HÔPITAL’S RULE
INDETERMINATE FORMS
IMPLICIT FUNCTIONS
LIMIT LAWS
yfx ax b xc==+ −()tan
2
• For evaluation of y, the inner function g(x) is determined first and then
substituted into the outer function f(x)
• Reversal of the order is not permitted (fro instance, f(g) πg(f))
yfgx x gx x fx gx
yfgx xa gx xa fx gx
yfgx x gx x fx gx
y
=
[]
=− ⇒ = =
[]
=
[]
=−=− =
=
[]
=−=− =
[]
() ( ) () () ()
() sin( ) () () sin ()
() ln () ln () ()
//
/
22
11
1313
12
But: == is a of two functionsxxfx gx product=•sin ( ) ( )
lim ()
()
xa
fx
gx
=
0
0 or
lim ()
() lim ()
() lim
()
()
()
()
xa xa xa
fx
gx
fx
gx
fnx
gnx
→→ →
=
lim ( ) lim[ ] lim[ ( ) ( )] lim ( ) lim ( )
lim lim( ) limsin lim( )
ln ( )
/
xa xa
fx
xa xa xa
n
n
x
x
xx
x
fx e fx gx fx gx
nxe ax
xax
en
→→ → →
→∞ → →
=⋅=
+
+= = =
=
1111
0
1
00
( = integer)
lim ( ) ( )fxL fxL xa
xa
=
εδ
where , = arbitrarily small numbers
()ffx L x a
x
−< <−<
ε
xa
x
<−<
δ
0
fx
if x
fx
if x
x
() ,
()
=
=1,
<
Discontinuity
at =0
1
0
0
fx x
fx
x
() /
=−123
No discontinuity
in ( )
Discontinuity in
tangent at
lim
lim sin
xx
x
x
x
x
x
+
()
=
+
()
()
2
211
2
21
4
But: lim
x0
does not exist because 1/xincreases
indefinitely
does not exist because sin xoscillates
indefinitely between ± 1
y = Dependent variable
x = Independent variable
f(x) = Explicit function of x
a, b, c = Constants
Fxy x yx y(,) sin=−+=
3120
• Equation cannot be easily solved for one variable
• Operations are conducted on the implicit form directly
COMPOSITE FUNCTIONS: FUNCTIONS OF A FUNCTION
• The differentiable functions f(x) and g(x) vanish at x= aso that
f(a)/g(a) = 0/0, provided the limit on
the right either exists or is ±
and g’(x) 0 for x=a
lim ()
() lim ()
()
xa
fx
gx xa
f' x
g' x
=
• The limits to 0/0, /, , 0 ¥ , 00, 0, and 1 are obtained by
expressions of L’Hôpital’s Rule according to the following procedures
Type 1 Type 2
Type 3 Type 4
• Limit found by repeated
differentiation
• Use this identity to reduce to
0/0 or /
fx gx fx
gx
gx
fx
() () ()
()
()
()
⋅= =
11
lim ( ) ( )
xa
fx gx
⋅=0
Factor or find common
denominator to reduce to 0/0
or /
lim [ ( ) ( )]
xa
fx gx
−=
lim ( ) () ,,
xa
fxgx
=∞
000
1
Use following transformation
to reduce to 0•:
y= f(x)g(x) ÆIny=
g(x) In f(x) Æ0•
FREQUENTLY USED LIMITS
• Equation is explicitly
solved for one of the
variables
• Limits arise in defining derivatives and integrals, in
establishing asymptotes to curves, in the evaluation of
indeterminate forms, and in establishing convergence of
sequences and series
• If a function f(x) increasingly approaches a constant value
Las xÆaand thereafter its distance from Lremains
arbitrarily small no matter how close the approach of xto
a,Lis the limit of f(x); this is mathematically expressed as
follows:
• Used to derive expressions (such as derivatives, integrals)
Addition Law: Limit of a sum
equals the sum of the limits
Product Law: Limit of
a product equals product
of the limits
Quotient Law: Limit of
a quotient equals quotient
of the limits, provided
the limit of the denominator
is not zero
Reciprocal Law:
Substitution Law
Composite Functions:
Squeeze Law: If a function g(x) is
positioned between two functions
f(x) and h(x) with the common
limit L, it will tend to the same limit L
The Number e:
Continuity of Functions:
A function f(x) is continuous
in an interval [a,b] if it does not
experience a jump or tend to ± x(namely, a single limit exists
at each point of the interval, including the end points)
lim ( ) ( ) lim ( ) lim ( )
xa xa xa
fx gx fx gx
→→
±= ±
[]
lim ( ) ( ) lim ( ) lim ( )
xa xa xa
fx gx fx gx
→→
•= •
[]
lim ()
()
lim ( )
lim ( ) lim ( )
xa
xa
xa
xa
fx
gx
fx
gx gx
=≠0
If then provided lim ( ) , lim ()
xa xa
fx L fx L L
→→
==
11 0
If and
then
lim ( ) , lim ( ) ( )
lim ( ) lim ( ) ( )
xa xa
xa xa
gx L f x fL
fgx f gx fL
→→
→→
==
==
[]
If
then
lim ( ) lim ( ) ,
lim ( )
() () ()
xa xa
xa
fx hx L
gx L
fx gx hx
→→
==
=
≤≤
The limit of the expression (1+ )
1/h is given by
lim
h0
1/h ... = e = 2.71828
h
h
+=++++( ) !!!
11
1
1
1
2
1
3
lim ( ) ( )
xc
fx fc a c b
=≤ for
Cusp
TM
permacharts
DIFFERENTIAL CALCULUS • A-763-3 1www.permacharts.com