Calculus Of One Real Variable By Pheng Kim Ving

1.1.4 
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1. OneSided Limits 
and N, contradicting the uniqueness of
limits; see Section
1.1.2 Theorem 2.1. However, although the limit of f at b
doesn't exist, its limits where only one side of b is taken into
account exist. This situation has led to the concept of
onesided limits.

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2. LeftHand Limits 
We say that a function f has the lefthand limit M as x approaches a, or that f
approaches M as x approaches a from
the left, and we write:
if the domain of f
contains points arbitrary close to but smaller than a and f(x) approaches arbitrarily close to L as x
approaches a from
the left (x
increases toward a).
In Fig. 1.1,
f has a lefthand limit of L at x = a
and also a lefthandlimit of M at x = b,
so that:
As x approaches 2 from the left, it's clear that x^{3} approaches 8
and thus x^{3}
1 approaches 7. The evaluation of
lefthand
limits is the same as that of (twosided) limits; see Section
1.1.2. Note that we say x approaches a from the left , but we
don't say f(x) approaches L from the left or
below; we don't care which side f(x) approaches L from.
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3. RightHand Limits 
We say that a function f
has the righthand limit L
as x approaches a, or that f
approaches L as x approaches
a from
the right, and we write:
if the domain of f
contains points arbitrary close to but greater than a and f(x) approaches arbitrarily close to L as x
approaches a from
the right (x
decreases toward a).
In Fig. 1.1,
f has a righthand limit of L at x = a
and also a righthandlimit of N at x = b,
so that:
As x approaches 2 from the right, it's clear that x^{3} approaches 8
and thus x^{3}
1 approaches 7. The evaluation of
righthand limits is the same as that of (twosided) limits; see Section
1.1.2. Note that we say x approaches a from the
right , but we
don't say f(x) approaches L from the right or
above; we don't care which side f(x) approaches L from.
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4. Evaluation 
Right and lefthand limits are referred to
as onesided limits. As seen in Examples 2.1
and 3.1, the evaluation of
onesided limits is the same as that of (twosided) limits; see Section
1.1.2.
Note that we say x
approaches a from the right or x approaches a from the left , but we don't say f(x)
approaches L
from the right or above or f(x) approaches L from the left or
below ; we
don't care which side f(x) approaches L from
or if it does so from both sides of L.
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5. Limits And OneSided Limits 
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6. Piecewise Functions 
Example 6.1
Let:
Solution
1.

Fig. 6.1 Graph Of y = f(x). 
EOS
Piecewise Functions
A piecewise function, also called a casedefined function, is a function whose graph consists of 2 or more pieces defined by different formulas, or by a single rule whose implementation changes with some subsets of the domain of the function, like the step function discussed in Problem & Solution 6. Direct substitution for limits, as discussed in Section 1.1.2 Part 4, isn't applicable to piecewise functions, as illustrated in Example 6.1 above.
A
point of formula change is a point where the function changes formula. In this
example, the points of formula change are x = 6, 2, and 3. To find the limit of a piecewise
function at a point of formula change, we must consider both onesided limits. This
is because the formulas are different on each side. For a more subtle case of
piecewise functions see Problem & Solution 6.
This example clearly demonstrates that the limit of a piecewise function at a point of formula change may or may not exist, and if it exists it may or may not be equal to the value of the function at that point if that value exists. So direct substitution for limits isn't applicable to piecewise functions at such points. To find limits at such points we must consider both onesided limits, as previously stated.
Problems & Solutions

1. Determine:
_{}
Solution
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2. Find:
_{}
if it exists.
Solution
If x < 2, then x 2 < 0, so x 2 = (x 2); thus:
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3. Find:
_{}
if it exists.
Solution
From problem & solution 2 we get:_{}
5. Let:
a. 

Graph
Of y = f(x). 
6. Let [x] denote the greatest integer less
than or equal to x.
a. Sketch a graph of [x]. Because of the look of its graph,
this function is called a step function
and the notation [x]
can be read step of x, or, for short, step x.
a.