Calculus Of One Real Variable By Pheng Kim Ving
Chapter 1: Limits And Continuity Section 1.1.3: The Indeterminate Form Of Type 0/0

 

1.1.3

The Indeterminate Form Of Type 0/0

 

 

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1. The Indeterminate Form Of Type 0/0

 

Example 1.1

 

Find this limit if it exists:

 

 

Note

 

We cannot use Theorem 3.2 iv of Section 1.1.2 because the denominator here approaches 0. If we tried then we'd get
0/0, the numerator also approaching 0. Since we cannot divide anything including 0 by 0, we don't know whether or not
this limit exists or what it is if it exists. We however can use a calculator to find out what it might be, as done in Fig. 1.1
as follows:

 

Fig. 1.1

 

 

We see that the given limit appears to exist and equal 10. We can find it formally as follows.

 

Solution

EOS

 

Indeed the limit is 10.

 

The Indeterminate Form 0/0

 

A rational function is a ratio or fraction of two polynomials. Let's examine the limit of a rational function when both its
numerator and denominator approach 0. Such a limit is said to be in the form 0/0. Let's consider the following three
trivial limits:

 

 

When both the numerator f(x) and denominator g(x) of a rational function r(x) = f(x)/g(x) approach 0 as x approaches
some point a, the limit of r is said to be of the indeterminate form of type 0/0.

 

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2. Not A Real Fraction

 

The form 0/0 describes that both the numerator and denominator of r approach 0, at either the same or different speeds,
as
x approaches a. It doesn't say that the limit of r equals 0/0 one cannot divide anything by 0. Remember, x remains
different from
a, so the denominator remains different from 0. The form 0/0 is a descriptive form, not a real fraction.

 

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3. Cancellation Of The Common Factor

 

When a limit is of the indeterminate form of type 0/0, you must do some algebraic manipulation to get rid of that form.

One method is to factor both the numerator and denominator, and then cancel out the common factor, as done in the
solution of example 1.1 and for other limits in Part 1. Remark that in solution of example 1.1 we can cancel out

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Problems & Solutions

 

1. Find this limit if it exists:

 

 

Solution

 

 

 

2. Find this limit if it exists:

 

 

Solution

 

 

 

3. Find:



if it exists.

 

Solution

 

 

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4. Does:



exist
?

 

Solution

 



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5. Evaluate:



if it exists.

 

Solution

 

 

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