Limits Evaluation by immediate or delayed substitution
Immediate substitution of x = a into an expression f(x) is possible
when and only when a the expression or function f(x) is continuous at x
= a. When that is not the case, one attempts to cancel factors and
divisors to arrive at a formula or function that continuous at x=a,
whence after that work or delay, limit evaluation by immediate
substitution is possible.
Here are some limits we are going to evaluate algebraically using the
algebraically described, numerical properties of limits - those described
in the previous lesson.
A = lim x->
2 3x+4
B = lim x->
6 5 x2- 8x
C = lim x->
-2 4x3-3x+ 1
D = lim y ->
-2 4y3-3y+ 1
and
Solutions
For the calculation
A = lim x->
2 3x+4
as x -> 2 we observe or assume 3x
-> 6 and so observe or assume 3x+4
-> 10. We can also write more
briefly
A = lim x->
2 3x+4 = 6+ 4 = 10 (result)
Second
B = lim x->
6 5 x2- 8x
= 5(6)2 - 8(6) = 5(30) - 48
= 150 - 48 = 102 (result)
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The limit evaluation process x-> 6 applied to 5 x2-
8x, an expression dependent on x, results in a number -25 which
does not depend on x. This limit evaluation process eliminates
the x dependence. When we apply a limit process to an formula or
function f(x) which eliminates the x-dependence, we call the
letter or placeholder x, a dummy variable.
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Third
C = lim x->
-2 4x3-3x+ 1
= 4 (-2)3 - 3(-2) + 1
= 4(-8) +6 + 1
= -32+ 7
= -25 (result)
Fourth we can evaluate
D = lim y ->
-2 4y3-3y+ 1 = -25
directly by same reasoning we did for C. Simply replace the x by a y.
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More on Dummy Variables: The letters x and
y in the expressions for C and D have the same roles. In the
expressions x-> -2 and y
-> -2 they both represent the ideas
of a number approaching the value -2. But the results for C and D
do not depend on our choice of letters in the limit expressions
for them.
In the evaluation of a limit
the value L of the limit does not depend on x,
as limit evaluation eliminates that dependence, but the value
of L may depend on a.
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Fifth, for the evaluation of the limit
we observe the attempt to evaluate inside expression
by the immediate substitution of 5 for x (x =5) in the yields 0/0, a
fraction with a zero in the denominator, a fraction which has no
numerical definition or value. It is undefined. But observe the inside
expression
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f(x) =
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x-5
x2-5x
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=
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x-5
(x-5)x
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=
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1
x
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->
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1
5
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when x -> 5
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The foregoing suggests the values of the inside expression f(x) =
x-5
x2-5x
approach 1/5 or 0.2 as x approaches 5 even thought f(0) is not defined.
Note: The nuance, subtlely or technicality here is that in the
evaluation of a limit
the value of the inside expression f(x) at the limiting value x = a
of x as x approaches a is not of interest. It does not have to be
defined. Limit evaluation here is independent of what happens at x
= a. Limit evaluation is based gives the limiting value of f(x)
when x is restricted to smaller and smaller a-deleted intervals
centred at x = a, that is intervals in which the value a has been
removed.
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The foregoing discussion needs to be understood, but when we evaluate the
limit
we write less. In particular we write
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E =
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lim
x->
5
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x-5
x2-5x
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=
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lim
x->
5
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x-5
(x-5)x
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=
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lim
x->
5
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1
x
|
|
=
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1
5
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(exact
answer)
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The avoidance of 0/0 by replacing the initial expression by another
represents a delay substitution.
Leave your results as a fractions. Here the
answer can be expressed exactly as a decimal but for instructors or
markers in correcting your results and any work leading to your
results,it easier to recognize a fraction (reduced to lowest forms)
than it is to recognize a decimal. So do exact arithmetic with
fractions and radicals (square roots etc instead of using your
calculator. This instructionto do exact arithmetic and avoid decimals,
or postpone their use until all possible exact arithmetic is done with
whole numbers and fractions provides a standard to meet in your mastery
of calculus.
The limit evaluation process may be written over several lines with
vertical alignment of equal signs - recommended for legibility. You could
do it in fewer lines with vertical alignment of equal signs optional, but
still advised for readability.
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