Calculus Of One Real Variable By Pheng Kim Ving
Chapter 1: Limits And Continuity Section 1.1.5: Limits At Infinity And Infinite Limits

 

1.1.5
Limits At Infinity And Infinite Limits

 


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1. Limits At Infinity

 

 

 

 

 

 

The formal definitions of limits at infinity are stated as follows:

 

 

Example 1.1

 

Find this limit:

 

 

Solution

EOS

 

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2. Horizontal Asymptotes


 

Fig. 2.1

Line
y = L is a horizontal asymptote of f.

 

Example 2.1

 

Find horizontal asymptotes of f(x) = 1/x.

 

Solution

EOS

 

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3. Infinite Limits

 

As examples we have:


 

 

 

 

Fig. 3.1

Line
x = a is a vertical asymptote of f.

 

Some other examples:

 


are infinite limits. An infinite limit may be produced by having the independent variable approach a finite point or infinity.

 

 

Note this distinction: a limit at infinity is one where the variable approaches infinity or negative infinity, while an infinite
limit is one where the function approaches infinity or negative infinity (the limit is infinite).

 


The formal definitions of infinite limits are stated as follows:

 

 

Example 3.1

 

Find this limit:

 

 

Solution

EOS

 


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4. Vertical Asymptotes


If:

 

respectively. See Fig. 3.1. The vertical line x = a is called a vertical asymptote of f. Remark that the line x = a can
be a vertical asymptote of
f only if f isn't defined at the point x = a.

 

Example 4.1

 

Find vertical asymptotes of f(x) = 1/x.

 

Solution

EOS


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Problems & Solutions


1. Find each of the following limits if it exists. Specify any horizontal or vertical asymptotes of the graphs of the functions.




Solution

 

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2. Find the following limit if it exists. Specify any horizontal or vertical asymptotes of the graph of the function.




Solution

 

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3. Let:

 



Solution

 

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4. Let:

 



Solution

 

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5. Find any horizontal and vertical asymptotes of each of the following functions.




Solution

 

 

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