Section 1.1 An Introduction To Limits
x | sin(x)/x |
0.9 | 0.870363 |
0.99 | 0.844471 |
0.999 | 0.841772 |
1 | 0.841471 |
1.001 | 0.841170 |
1.01 | 0.838447 |
1.1 | 0.810189 |
x | sin(x)/x |
-0.1 | 0.9983341665 |
-0.01 | 0.9999833334 |
-0.001 | 0.9999998333 |
0 | not defined |
0.001 | 0.9999998333 |
0.01 | 0.9999833334 |
0.1 | 0.9983341665 |
Example 1.1.8. Approximating the value of a limit.
Use graphical and numerical methods to approximate
To graphically approximate the limit, graph
on a small interval that contains \(3\text{.}\) To numerically approximate the limit, create a table of values where the \(x\) values are near \(3\text{.}\) This is done in Figure 1.1.9 and Figure 1.1.10, respectively.
\(x\) | \(\frac{x^2-x-6}{6x^2-19x+3}\) |
\(2.9\) | \(0.29878\) |
\(2.99\) | \(0.294569\) |
\(2.999\) | \(0.294163\) |
3 | not defined |
\(3.001\) | \(0.294073\) |
\(3.01\) | \(0.293669\) |
\(3.1\) | \(0.289773\) |
The graph shows that when \(x\) is near \(3\text{,}\) the value of \(y\) is very near \(0.3\text{.}\) By considering values of \(x\) near \(3\text{,}\) we see that \(y=0.294\) is a better approximation. The graph and the table imply that
If a graph does not produce as good an approximation as a table, why bother with it?
How many values of x in a table are βenough?β In the previous example, could we have just used x=3.001 and found a fine approximation?
Example 1.1.11. Approximating the value of a limit.
Graphically and numerically approximate the limit of f(x) as x approaches 0, where
Again we graph \(f(x)\) and create a table of its values near \(x=0\) to approximate the limit. Note that this is a piecewise defined function, so it behaves differently on either side of \(0\text{.}\) Figure 1.1.12 shows a graph of \(f(x)\text{,}\) and on either side of \(0\) it seems the \(y\) values approach \(1\text{.}\) Note that \(f(0)\) is not actually defined, as indicated in the graph with the open circle.
\(x\) | \(f(x)\) |
\(-0.1\) | \(0.9\) |
\(-0.01\) | \(0.99\) |
\(-0.001\) | \(0.999\) |
\(0.001\) | \(0.999999\) |
\(0.01\) | \(0.9999\) |
\(0.1\) | \(0.99\) |
Figure 1.1.13 shows values of \(f(x)\) for values of \(x\) near \(0\text{.}\) It is clear that as \(x\) takes on values very near \(0\text{,}\) \(f(x)\) takes on values very near \(1\text{.}\) It turns out that if we let \(x=0\) for either βpieceβ of \(f(x)\text{,}\) \(1\) is returned; this is significant and we'll return to this idea later.
The graph and table allow us to say that \(\lim_{x\to 0}f(x) \approx 1\text{;}\) in fact, we are probably very sure it equals 1.
Subsection 1.1.1 Identifying When Limits Do Not Exist
The function f(x) may approach different values on either side of c.
The function may grow without upper or lower bound as x approaches c.
The function may oscillate as x approaches c without approaching a specific value.
Example 1.1.15. Different Values Approached From Left and Right.
Explore why limxβ1f(x) does not exist, where
A graph of \(f(x)\) around \(x=1\) and a table are given in Figures Figure 1.1.16 and Figure 1.1.17, respectively. It is clear that as \(x\) approaches \(1\text{,}\) \(f(x)\) does not seem to approach a single number. Instead, it seems as though \(f(x)\) approaches two different numbers. When considering values of \(x\) less than \(1\) (approaching \(1\) from the left), it seems that \(f(x)\) is approaching \(2\text{;}\) when considering values of \(x\) greater than \(1\) (approaching \(1\) from the right), it seems that \(f(x)\) is approaching \(1\text{.}\) Recognizing this behavior is important; we'll study this in greater depth later. Right now, it suffices to say that the limit does not exist since \(f(x)\) is approaching two different values as \(x\) approaches \(1\text{.}\)
\(x\) | \(f(x)\) |
\(0.9\) | \(2.01\) |
\(0.99\) | \(2.0001\) |
\(0.999\) | \(2.000001\) |
\(1.001\) | \(1.001\) |
\(1.01\) | \(1.01\) |
\(1.1\) | \(1.1\) |
Example 1.1.18. The Function Grows Without Bound.
Explore why limxβ11(xβ1)2 does not exist.
A graph and table of \(f(x) = \frac{1}{(x-1)^2}\) are given in Figure 1.1.19 and Figure 1.1.20, respectively. Both show that as \(x\) approaches \(1\text{,}\) \(f(x)\) grows larger and larger.
\(x\) | \(f(x)\) |
\(0.9\) | \(100\text{.}\) |
\(0.99\) | \(10000\text{.}\) |
\(0.999\) | \(1.\times 10^6\) |
\(1.001\) | \(1.\times 10^6\) |
\(1.01\) | \(10000\text{.}\) |
\(1.1\) | \(100\text{.}\) |
We can deduce this on our own, without the aid of the graph and table. If \(x\) is near 1, then \((x-1)^2\) is very small, and:
Since \(f(x)\) is not approaching a single number, we conclude that
does not exist.
Example 1.1.22. The Function Oscillates.
Explore why limxβ0sin(1/x) does not exist.
Two graphs of \(f(x) = \sin(1/x)\) are given in Figure 1.1.23. Figure 1.1.23.(a) shows \(f(x)\) on the interval \([-1,1]\text{;}\) notice how \(f(x)\) seems to oscillate near \(x=0\text{.}\) One might think that despite the oscillation, as \(x\) approaches \(0\text{,}\) \(f(x)\) approaches \(0\text{.}\) However, Figure 1.1.23.(b) zooms in on \(\sin(1/x)\text{,}\) on the interval \([-0.1,0.1]\text{.}\) Here the oscillation is even more pronounced. Finally, in Figure 1.1.24, we see \(\sin(1/x)\) evaluated for values of \(x\) near \(0\text{.}\) As \(x\) approaches \(0\text{,}\) \(f(x)\) does not appear to approach any value.
\(x\) | \(\sin(1/x)\) |
\(0.1\) | \(-0.544021\) |
\(0.01\) | \(-0.506366\) |
\(0.001\) | \(0.82688\) |
\(0.0001\) | \(-0.305614\) |
\(1.\times 10^{-5}\) | \(0.0357488\) |
\(1.\times 10^{-6}\) | \(-0.349994\) |
\(1.\times 10^{-7}\) | \(0.420548\) |
It can be shown that in reality, as \(x\) approaches 0, \(\sin(1/x)\) takes on all values between \(-1\) and \(1\) infinitely many times! Because of this oscillation, \(\lim_{x\to 0}\sin(1/x)\) does not exist.
Subsection 1.1.2 Limits of Difference Quotients
We have approximated limits of functions as x approached a particular number. We will consider another important kind of limit after explaining a few key ideas.Since the particle traveled 10 feet in 4 seconds, we can say the particle's average velocity was 2.5 ft/s. We write this calculation using a βquotient of differences,β or, a difference quotient:f(1)=10 and f(5)=20.
h | f(1+h)βf(1)h |
β0.5 | 9.25 |
β0.1 | 8.65 |
β0.01 | 8.515 |
0.01 | 8.485 |
0.1 | 8.35 |
0.5 | 7.75 |
Exercises 1.1.3 Exercises
Terms and Concepts
1.
In your own words, what does it mean to βfind the limit of f(x) as x approaches 3β?
2.
An expression of the form 00 is called .
3.
True
False
4.
Describe three situations where limxβcf(x) does not exist.
5.
In your own words, what is a difference quotient?
6.
When x is near 0, sinxx is near what value?
Problems
Approximate the limit numerically and graphically.
7.
limxβ1(x2+3xβ5)
8.
limxβ0(x3β3x2+xβ5)
9.
limxβ0(x+1x2+3x)
10.
limxβ3(x2β2xβ3x2β4x+3)
11.
limxββ1(x2+8x+7x2+6x+5)
12.
limxβ2(x2+7xβ10x2β4x+4)
13.
limxβ2f(x), where f(x)={x+2if xβ€23xβ5if x>2
14.
limxβ3f(x), where f(x)={x2βx+1if xβ€32x+1if x>3
15.
limxβ0f(x), where f(x)={cos(x)if xβ€0x2+3x+1if x>0
16.
limxβΟ2f(x), where f(x)={sin(x)xβ€Ο2cos(x)x>Ο2
17.
limxβ0|x|x
18.
limxβ0eβe1/x
19.
limxββ5β|x|β!, where |x| is the absolute value of x, βxβ is the floor of x (the greatest integer less than or equal to x), and x! is x factorial.
20.
limxββ1β|x|β!, where |x| is the absolute value of x, βxβ is the floor of x (the greatest integer less than or equal to x), and x! is x factorial.
Approximate the limit of the difference quotient, limhβ0f(a+h)βf(a)h, using h=Β±0.1,Β±0.01.