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10 Introduction to theDerivative
Copyright © Cengage Learning. All rights reserved.
10.3 Limits and Continuity: Algebraic Approach
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Limits and Continuity: Algebraic Approach
Closed-Form Functions
A function is written in closed form if it is specified by combining constants, powers of x, exponential functions, radicals, logarithms, absolute values, trigonometric functions (and some other functions we do not encounter in this text) into a single mathematical formula by means of the usual arithmetic operations and composition of functions.
A closed-form function is any function that can be written in closed form.
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Limits and Continuity: Algebraic Approach
Quick Examples
1. 3x2 – | x | + 1, and are
written in closed form, so they are all closed-form functions.
–1 if x ≤ –1
2. f (x) = x2 + x if –1 < x ≤ 1
2 – x if 1 < x ≤ 2
is not written in closed-form because f (x) is not expressed by a single mathematical formula.
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Limits and Continuity: Algebraic Approach
Theorem 3.1 Continuity of Closed-Form Functions
Every closed-form function is continuous on its domain. Thus, if f is a closed-form function and f (a) is defined, we have limx→a f (x) = f (a).
Quick Example
f (x) = 1/x is a closed-form function, and its natural domain consists of all real numbers except 0. Thus, f is continuous at every nonzero real number.
That is,
provided a ≠ 0.
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Example 1 – Limit of a Closed-Form Function at a Point in Its Domain
Evaluate algebraically.
Solution:
First, notice that (x3 – 8)/(x – 2) is a closed-form function because it is specified by a single algebraic formula.
Also, x = 1 is in the domain of this function.
Therefore,
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Limits and Continuity: Algebraic Approach
Functions with Equal Limits
If f (x) = g(x) for all x except possibly x = a, then
Quick Example
for all x except x = 1.
Therefore,
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Limits at Infinity
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Example 5 – Limits at Infinity
Compute the following limits, if they exist:
1010
Example 5 – Solution
Solution for a and b.
The highest power of x in both the numerator and
denominator dominate the calculations.
For instance, when x = 100,000, the term 2x2 in the numerator has the value of 20,000,000,000, whereas the term 4x has the comparatively insignificant value of 400,000.
Similarly, the term x2 in the denominator overwhelms the term –1.
1111
Example 5 – Solution
In other words, for large values of x (or negative values with large magnitude),
Therefore,
cont’d
Use only the highest powers top and bottom.
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Example 5 – Solution
c. Applying the previous technique of looking only at highest powers gives
As x gets large, –x/ 2 gets large in magnitude but negative, so the limit is
cont’d
Use only the highest powers top and bottom.
Simplify.
1313
Example 5 – Solution
d.
As x gets large, 2/(5x) gets close to zero, so the limit is
cont’d
Use only the highest powers top and bottom.
1414
Example 5 – Solution
e. Here we do not have a ratio of polynomials. However, we know that, as t becomes large and positive, so does e0.1t, and hence also e0.1t – 20.
Thus,
cont’d
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Example 5 – Solution
f. As t → , the term (3.68)–t = in the denominator,
being 1 divided by a very large number, approaches zero.
Hence the denominator 1 + 2.2(3.68)–t approaches
1 + 2.2(0) = 1 as t → .
Thus,
cont’d
1616
Limits at Infinity
Theorem 3.2 Evaluating the Limit of a Rational Function at
If f (x) has the form
with the ci and di constants (cn 0 and dm 0), then we can calculate the limit of f (x) as x → by ignoring all powers of x except the highest in both the numerator and denominator. Thus,
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Limits at Infinity
Quick Example
1818
Some Determinate and Indeterminate Forms
1919
Some Determinate and Indeterminate Forms
Some Determinate and Indeterminate Forms
are indeterminate; evaluating limits in which
these arise requires simplification or further analysis.
The following are determinate forms for any nonzero number k:
2020
Some Determinate and Indeterminate Forms
and, if k is positive, then
2121
Some Determinate and Indeterminate Forms
Quick Examples
1.
2.
3.
