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

To explain: The meaning of limx→af(x)=L.

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:

To explain: The meaning of limx→a+f(x)=L.

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:

To explain: The meaning of limx→a−f(x)=L.

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:

To explain: The meaning of limx→af(x)=∞.

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:

To explain: The meaning of limx→∞f(x)=L.

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: