NOTE: You may view the original PDFs of the notes written in class by clicking on the link for the 1:30 Section or 2:30 Section, respectively.
You will need a calculator today. Examples of what buttons you need to press on the TI-30 Xa for various calculations can be found here.
Lecture 2: Introduction to Limits
GOAL: Gain an intuitive understanding of
what it means for one value to approach another.
Compare these two functions at \( x = 3 \)
Figure 1
Two hand-drawn coordinate sketches side by side.
The left graph shows a straight line through the
origin labeled \( f(x) = 2x \), with a filled
dot at the point where \( x = 3 \) and the
y-value 6 marked on the vertical axis; red
arrows point toward the filled dot from both
sides along the line. Beside it in red are the
notes \( f(3) = 6 \) and \( \lim_{x \to 3} f(x)
= 6 \). The right graph shows a straight line
through the origin labeled \( g(x) =
2x\frac{(x-3)}{(x-3)} \), with an open circle
(hole) at the point where \( x = 3 \) and the
value 6 marked on the vertical axis. Beside it
in red are the notes \( g(3) \) undefined and \(
\lim_{x \to 3} g(x) = 6 \). On both graphs the
tick mark 3 is labeled on the horizontal axis.
We say x approaches a (written \( x \to a \)) to
mean "\( x \) gets really really close to \( a \)." So in Figure 1, as \( x \to 3 \), we have \( f(x) \to 6 \) and \( g(x) \to 6 \)
We write \( \lim\limits_{x \to a} f(x) = L \)
as shorthand for "as \( x \to a \), \( f(x) \to L \)." \( L
\) is called the limit of \( f(x) \) as \( x \to a
\).
When dealing with limits, what happens at \(a\) is not important. Rather, we want to know what value \(f(x)\) gets close to as \(x \to a\). So, in Figure 1, even though \(f(3) = 6\) and \(g(x)\) is undefined.
\[
\lim_{x \to 3} f(x) = 6\quad\text{and}\quad\lim_{x \to 3} g(x) = 6
\]
Example 1
Find a plausible value for \( \lim\limits_{x \to 4} \left(
2\sqrt{x} - 1 \right) = 3 \)
Values of \( f(x) = 2\sqrt{x} - 1 \) near \( x = 4
\)
\( x \)
3.9
3.99
3.999
4
4.001
4.01
4.1
\( f(x) \)
2.9497
2.995
2.9995
3.0005
3.005
3.0497
To compute \( 2\sqrt{4.1} - 1 \) on our calculator, click the following buttons in this order: 4.1 \( \to \)
\( \sqrt{x} \) \( \to \)
\( \times \) \( \to \)
2 \( \to \) = \( \to
\) - \( \to \) 1 \(
\to \) =
A hand-drawn coordinate graph of the
Heaviside step function \( H(x) \). A
vertical axis and a horizontal axis cross at
the origin. For \( x \ge 0 \) a horizontal
ray is drawn at height 1, starting at a
filled dot on the vertical axis at height 1
(labelled 1) and extending to the right with
an arrowhead. For \( x < 0 \) a horizontal
ray lies along the horizontal axis at height
0, starting at an open circle at the origin
and extending to the left with an arrowhead,
showing the jump discontinuity at \( x = 0
\).
Values of \( H(x) \) near \( x = 0 \)
\( x \)
-0.1
-0.01
-0.001
0
0.001
0.01
0.1
\( H(x) \)
0
0
0
1
1
1
\( H(x) \) will not approach a particular value, so we
say \( \lim\limits_{x \to 0} H(x) \) does not exist (DNE).
Question: Can we be more specific as to why it DNE?
Answer: There are two tools to help us with this, one-sided limits and infinite limits.
One-Sided Limits
We write \( \lim\limits_{x \to a^-} f(x) = L \) to mean "\( f(x)
\to L \) as \( x \to a \) from the left." From Example 3,
\[ \lim_{x \to 0^-} H(x) = 0 \]
Similarly, \( \lim\limits_{x \to a^+} f(x) = L \) means "\( f(x)
\to L \) as \( x \to a \) from the right." From Example 3,
\[ \lim_{x \to 0^+} H(x) = 1 \]
Remark
If these are unequal, then the limit DNE.
If they are equal, then \[ \lim_{x \to a} f(x) = \lim_{x
\to a^-} f(x) = \lim_{x \to a^+} f(x) \]
Example 4 (Highlighting (2) in the Remark)
Compute \( \lim_{x \to 0} |x| \)
Figure 3
A hand-drawn coordinate plane in red with a
horizontal x-axis and a vertical y-axis
crossing at the origin. The graph of \( |x|
\) is drawn as a V shape with its vertex at
the origin, rising to the left and to the
right. The curve is labeled \( |x| \) at its
upper right. Blue arrows are drawn along
both branches pointing inward and downward
toward the origin, indicating that the
function values approach 0 as \( x \)
approaches 0 from both the left and the
right.
A hand-drawn graph of \( \frac{1}{x^2} \) on
a coordinate plane with a horizontal axis
and a vertical axis. Two branches, one on
each side of the vertical axis, are low and
nearly flat far from the origin and curve
sharply upward as they approach the vertical
axis. Two thick red arrows point upward
along the two branches near the vertical
axis, showing that the function increases
without bound as \( x \) approaches 0 from
either side. The curve is labeled \(
\frac{1}{x^2} \) to the right.
Values of \( f(x) = \frac{1}{x^2} \) near \( x = 0
\)
\( x \)
-0.1
-0.01
-0.001
0
0.001
0.01
0.1
\( f(x) \)
100
\( 10^4 \)
\( 10^6 \)
\( 10^6 \)
\( 10^4 \)
100
We say \( \lim_{x \to 0} \frac{1}{x^2} = \infty \). That
is, as \( x \to 0 \), \( f(x) \) keeps increasing w/o
bound.
Example 6
Figure 5
A hand-drawn graph of \( -\frac{1}{x^2} \)
on a coordinate plane. Two branches lie
below the horizontal axis, nearly flat far
from the origin and plunging steeply
downward as they approach the vertical axis
from the left and from the right. Two thick
red downward arrows are drawn along the
branches near the vertical axis, indicating
the function decreases without bound as \( x
\) approaches 0. The graph is labeled \(
-\frac{1}{x^2} \) to the right.
Figure 6
A hand-drawn graph of \( \frac{1}{x} \) on a
coordinate plane. The right branch lies
above the horizontal axis and rises steeply
upward as it approaches the vertical axis
from the right, marked with a thick red
upward arrow. The left branch lies below the
horizontal axis and plunges steeply downward
as it approaches the vertical axis from the
left, marked with a thick red downward
arrow. The curve is labeled \( \frac{1}{x}
\) at the right.
If the limit is \( \infty \) or \( -\infty \),
technically the limit DNE. BUT, we are being
more specific as to why it DNE.
Computing Limits Graphically
Figure 7
A hand-annotated coordinate grid from \( x = -2
\) to \( x = 3 \) and \( y = -3 \) to \( y = 3
\), showing a piecewise curve labeled f(x) in
black with red arrows tracing the direction of
approach. For \( x < -1 \) the graph is a
horizontal line at height 2 ending in a filled
dot at \( x = -1 \). At \( x = -1 \) there is an
open circle at height 1, from which a straight
segment descends to the origin, forming a V
shape with a second straight segment rising from
the origin to an open circle at \( (1, 1) \).
From \( x = 1 \) a curve rises steeply upward
toward infinity as \( x \) approaches the dashed
vertical asymptote at \( x = 2 \). To the right
of \( x = 2 \) another branch comes up from
negative infinity near the asymptote, rising
toward the right and labeled f(x). Red
arrowheads mark approach directions at \( x = -1
\), \( x = 0 \), \( x = 1 \), and on both sides
of the asymptote at \( x = 2 \).
