Explain what each of the following means and illustrate with sketch. (a) lim x → a f ( x ) = L (b) lim x → a + f ( x ) = L (c) lim x → a − f ( x ) = L (d) lim x → a f ( x ) = ∞ (e) lim x → ∞ f ( x ) = L
Explain what each of the following means and illustrate with sketch. (a) lim x → a f ( x ) = L (b) lim x → a + f ( x ) = L (c) lim x → a − f ( x ) = L (d) lim x → a f ( x ) = ∞ (e) lim x → ∞ f ( x ) = L
Explain what each of the following means and illustrate with sketch.
(a)
lim
x
→
a
f
(
x
)
=
L
(b)
lim
x
→
a
+
f
(
x
)
=
L
(c)
lim
x
→
a
−
f
(
x
)
=
L
(d)
lim
x
→
a
f
(
x
)
=
∞
(e)
lim
x
→
∞
f
(
x
)
=
L
(a)
Expert Solution
To determine
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)
Expert Solution
To determine
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)
Expert Solution
To determine
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)
Expert Solution
To determine
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)
Expert Solution
To determine
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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36
32
28
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16
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1
2
3
4
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Consider the following system of equations, Ax=b :
x+2y+3z - w = 2
2x4z2w = 3
-x+6y+17z7w = 0
-9x-2y+13z7w = -14
a. Find the solution to the system. Write it as a parametric equation. You can use a
computer to do the row reduction.
b. What is a geometric description of the solution? Explain how you know.
c. Write the solution in vector form?
d. What is the solution to the homogeneous system, Ax=0?
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