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Concept explainers
Explain what each of the following means and illustrate with sketch.
(a) limx→af(x)=L
(b) limx→a+f(x)=L
(c) limx→a−f(x)=L
(d) limx→af(x)=∞
(e) limx→∞f(x)=L
(a)
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To explain: The meaning of limx→af(x)=L.
Explanation of Solution
Result used:
Definition of limit:
Let f(x) be a function is defined when x approaches to p then limx→pf(x)=L, if for every number ε>0 there is some δ>0 such that |f(x)−L|<ε whenever 0<|x−p|<δ.
Graph:
Calculation:
The limit of the function limx→af(x)=L means the limit of f(x) equal to L when x approaches to a, if x is closer and closer to a from the both sides then the value of f(x) also closer and closer to L.
In the limit definition x≠a this means finding the limit of f(x) when x approaches to a, there no need to consider x=a.
There are three cases for define limx→af(x)=L.
Case (1):
The limit of the function limx→af(x)=L, if x approaches to a then the value of f(x) are closer to L and f(a) is L.
Graph:
Case (2):
The limit of the function limx→af(x)=L, if x approaches to a then the value of f(x) are closer to L and f(a) is undefined.
Case (3):
The limit of the function limx→af(x)=L, if x approaches to a then the value of f(x) are closer to other than L.
Graph:
(b)
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To explain: The meaning of limx→a+f(x)=L.
Explanation of Solution
Result used:
Definition of limit:
Let f(x) be a function is defined when x approaches to p then limx→pf(x)=L, if for every number ε>0 there is some δ>0 such that |f(x)−L|<ε whenever 0<|x−p|<δ.
Calculation:
limx→a+f(x)=L means the limit of f(x) equal to L when x approaches to a from the right, if x is closer and closer to a from the right and remains greater than a then the value of f(x) also closer and closer to L.
Graph:
(c)
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To explain: The meaning of limx→a−f(x)=L.
Explanation of Solution
Result used:
Definition of limit:
Let f(x) be a function is defined when x approaches to p then limx→pf(x)=L, if for every number ε>0 there is some δ>0 such that |f(x)−L|<ε whenever 0<|x−p|<δ.
Calculation:
The limit of the function limx→a−f(x)=L means the limit of f(x) equal to L when x approaches to a from the left, if x is closer and closer to a from the left and remains less than a then the value of f(x) also closer and closer to L.
Graph:
(d)
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To explain: The meaning of limx→af(x)=∞.
Explanation of Solution
Result used:
Definition of limit:
Let f(x) be a function is defined when x approaches to p then limx→pf(x)=L, if for every number ε>0 there is some δ>0 such that |f(x)−L|<ε whenever 0<|x−p|<δ.
Calculation:
The limit of the function limx→af(x)=∞ means the limit of f(x) is larger value when x approaches to a from the both sides. That is any M>0, f(x)>M for some x-value is sufficiently close to a.
Graph:
(e)

To explain: The meaning of limx→∞f(x)=L.
Explanation of Solution
Result used:
Definition of limit:
Let f(x) be a function is defined when x approaches to p then limx→pf(x)=L, if for every number ε>0 there is some δ>0 such that |f(x)−L|<ε whenever 0<|x−p|<δ.
Calculation:
The limit of the function limx→∞f(x)=L means the limit of f(x) is L when x approaches to larger value the graph get closer and closer to the line y=L.
Graph:
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