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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

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)

To determine

To explain: The meaning of limxaf(x)=L.

Explanation

Result used:

Definition of limit:

Let f(x) be a function is defined when x approaches to p then limxpf(x)=L, if for every number ε>0 there is some δ>0  such that |f(x)L|<ε whenever 0<|xp|<δ.

Graph:

images

Calculation:

The limit of the function limxaf(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 xa 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 limxaf(x)=L.

Case (1):

The limit of the function limxaf(x)=L, if x approaches to a then the value of f(x) are closer to L and f(a) is L.

Graph:

images

Case (2):

The limit of the function limxaf(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 limxaf(x)=L, if x approaches to a then the value of f(x) are closer to other than L.

Graph:

images

(b)

To determine

To explain: The meaning of limxa+f(x)=L.

Explanation

Result used:

Definition of limit:

Let f(x) be a function is defined when x approaches to p then limxpf(x)=L, if for every number ε>0 there is some δ>0  such that |f(x)L|<ε whenever 0<|xp|<δ.

Calculation:

limxa+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:

images

(c)

To determine

To explain: The meaning of limxaf(x)=L.

Explanation

Result used:

Definition of limit:

Let f(x) be a function is defined when x approaches to p then limxpf(x)=L, if for every number ε>0 there is some δ>0  such that |f(x)L|<ε whenever 0<|xp|<δ.

Calculation:

The limit of the function limxaf(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:

images

(d)

To determine

To explain: The meaning of limxaf(x)=.

Explanation

Result used:

Definition of limit:

Let f(x) be a function is defined when x approaches to p then limxpf(x)=L, if for every number ε>0 there is some δ>0  such that |f(x)L|<ε whenever 0<|xp|<δ.

Calculation:

The limit of the function limxaf(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:

images

(e)

To determine

To explain: The meaning of limxf(x)=L.

Explanation

Result used:

Definition of limit:

Let f(x) be a function is defined when x approaches to p then limxpf(x)=L, if for every number ε>0 there is some δ>0  such that |f(x)L|<ε whenever 0<|xp|<δ.

Calculation:

The limit of the function limxf(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:

images

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