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Calculus 1 Problems & Solutions – Chapter 1 – Section 1.1.2

1.1.2
Properties Of Limits

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Review

1. Absolute Values

The definition of the limit involves absolute values; see Section 1.1.1 Definition 2.1. Not surprisingly, the proofs of many
theorems about limits require the use of properties of absolute values. First let's recall the definition of absolute values.
We have |2| = 2 and |–2| = 2 = –(–2). To get the absolute value 2 of –2 we remove the minus sign, by multiplying –2 by
–1, or –1 by –2. The absolute value of 0 is 0: |0| = 0. So, the absolute value of any real number x is formally defined
as follows:

The properties of absolute values often used are:

where a and b are arbitrary real numbers and k is an arbitrary positive constant. Properties i and ii are obvious. Let's
consider property iii. For example:

|(2)(–3)| = |–6| = 6,
|2||–3| = (2)(3) = 6,
so that |(2)(–3)| = |2||–3|.

Next let's examine property iv. Two examples are:

|2 + 5| = |7| = 7,
|2| + |5| = 2 + 5 = 7,
so that |2 + 5| = |2| + |5|;

|2 + (–5)| = |–3| = 3,
|2| + |–5| = 2 + 5 = 7,
so that |2 + (–5)| < |2| + |–5|.

The property can be proved as follows:

ABC, as seen in Fig. 1.1. For any two points P and Q, denote the vector joining and directed from P to Q and its length

the reason for its name.

Fig. 1.1

Fig. 1.2

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2. Uniqueness Of Limits

Theorem 2.1

If a function f(x) has a limit L at a point x = a, then the limit L is unique.

Proof

which is a contradiction. Consequently, we must have M = L. That is, the limit of f(x) at x = a is unique.

EOP

Remark 2.1

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3. Properties Of Limits

The limits in i and ii and the implication in iii in the following theorem should be intuitively obvious.

Theorem 3.1

Proof

EOP

Again the following properties should be intuitively obvious.

Theorem 3.2

Proof

i. We have:

EOP

Remark 3.1

theorem 2.1, , so that you can see in later proofs why we can bypass it. Here and in the remainder of this tutorial we
bypass it and the like, as they do in textbooks.

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4. Direct Substitution

As an example we have:

	[by theorem 3.2 iv]
	[by theorem 3.2 ii]
	[by theorems 3.2 iii, 3.1 i, and 3.1 ii]
	[by theorem 3.1 ii]

We step-by-step apply the indicated theorems to evaluate the limit. However, in practice the evaluation isn't done in such
a step-by-step detail or with such explicit references to theorems. Instead it's done as follows.

Example 4.1

Find the following limit if it exists:

Solution

EOS

In this practical solution, we merely substitute the value x = 3 directly in the given function to get the value of the limit.

Now consider this limit:

In this section we deal only with limits that can be evaluated by direct substitution. This is the case where the limit of a function f(x) as x approaches a is obtained by substituting x = a directly in the given expression of f(x), which is possible by the properties of limits as stated in the theorems above.

The form 0/0 is called an indeterminate form. Limits for which direct substitution isn't applicable because it would lead to the form 0/0 or other indeterminate forms will be handled in later sections in this chapter and in Section 5.4.2. Direct substitution isn't applicable either to piecewise functions (also called case-defined functions) at points of formula change, as demonstrated in Section 1.1.4 Part 6.

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5. The Squeeze Theorem

For the Squeeze Theorem stated below, see Fig. 5.1.