10
Introduction to the
Derivative
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10.3
Limits and Continuity: Algebraic Approach
Copyright © Cengage Learning. All rights reserved.
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.
3
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.
4
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.
5
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,
6
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,
7
Limits at Infinity
8
Example 5 – Limits at Infinity
Compute the following limits, if they exist:
9
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.
10
Example 5 – Solution
cont’d
In other words, for large values of x (or negative values
with large magnitude),
Use only the highest powers top and bottom.
Therefore,
11
Example 5 – Solution
cont’d
c. Applying the previous technique of looking only at
highest powers gives
Use only the highest powers
top and bottom.
Simplify.
As x gets large, –x/2 gets large in magnitude but
negative, so the limit is
12
Example 5 – Solution
d.
cont’d
Use only the highest powers
top and bottom.
As x gets large, 2/(5x) gets close to zero, so the limit is
13
Example 5 – Solution
cont’d
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,
14
Example 5 – Solution
f. As t →
, the term (3.68)–t =
cont’d
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,
15
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,
16
Limits at Infinity
Quick Example
17
Some Determinate and
Indeterminate Forms
18
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:
19
Some Determinate and Indeterminate Forms
and, if k is positive, then
20
Some Determinate and Indeterminate Forms
Quick Examples
1.
2.
3.
21