Definition 1.1 (Intuitive Definition). The limit of f (x), as x approaches a, equals L means that as x gets arbitrarily close to the value a (but not actually equal to a), the value of f (x) gets close to the value L. This is also written
xlim→af (x) = L
Example 1.1. Consider the function f (x) = sin 2x5x and find limx→0f (x).
−2 −1 1 2
−0.1 0.1 0.2 0.3 0.4
x y
1
x sin(2x)5x x sin(2x)5x 0.5 0.336588394 -0.5 0.336588394 0.1 0.397338662 -0.1 0.397338662 0.001 0.399999733 0.001 0.399999733 0.0001 0.399999997 -0.001 0.399999997
0.00001 0.4 -0.0001 0.4
Example 1.2. Consider the function f (x) = sin16xπ . Use the table of values below to guess the value of limx→0f (x).
x sin(π/16x) x sin(π/16x)
0.5 0.382683432 -0.5 -0.382683432 0.1 0.923879533 -0.1 -0.923879533
0.001 1 0.001 -1
0.0001 2.57729E-13 -0.001 -2.57729E-13
Example 1.3. Consider the function f (x) = sin16xπ again. Use the table of values
below to guess the value of limx→0f (x).
n x = 1/(8(2n + 1)) sin(π/16x)
10 0.005952381 1
11 0.005952381 -1
1,000 6.24688E-05 1
1,001 6.24688E-05 -1
1,000,000 6.24688E-05 1
1,000,001 6.24688E-05 -1
−2 −1 1 2
−1
−0.5 0.5 1
x
y
Example 1.4. Consider the function f (x) = √ 3x−15
x2−10x+25 and find limx→5f (x).
n √ 3x−15
x2−10x+25 x √ 3x−15 x2−10x+25
5.5 3 4.5 -3
5.1 3 4.9 -3
5.01 3 4.99 -3
5.001 3.000000004 4.999 -2.999999998 5.0001 2.999999876 4.9999 -2.999999876
−5 5 10 15
−4
−2 2 4
x y
2. Left and Right Limits
Definition 2.1. The limit of f (x), as x approaches a from the left, equals L
lim
x→a+f (x) = L Theorem 2.1. lim
x→af (x) = L if and only if
Example 2.1. (2.2 WebAssign Homework (2.2.006)) The graph of y = f (x) is
−4 −2 2 4 6 8
−2 2 4 6
x
y (a) lim
x→−3−f (x) = (b) lim
x→−3+f (x) = (c) lim
x→−3f (x) = (d) f (−3) =
Example 2.2. Sketch a function with the following conditions: f (1) = 0, lim
x→1+f (x) =
−2, lim
x→1−f (x) = 2
3. Infinite Limits
Definition 3.1. The limit of f (x), as x approaches a is infinite means that as x gets arbitrarily close to the value a, the value of f (x) gets arbitrarily large. This is also written
x→alimf (x) =∞
If the value of |f(x)| gets arbitrarily large, but f(x) < 0, for x close to a, then we write
Example 4.1. (2.2 WebAssign Homework (2.2.009)) The graph of y = f (x) is
-7 -3 6
x y
(a) lim
x→−7f (x) = (f ) Equation of the
smallest vertical asymptote is
Example 4.2. Determine the limit. lim
x→0+ln x
Example 4.3. (2.2 WebAssign Homework (2.2.035)) Determine the limit. lim
x→1+ln(x2− 1)
Example 4.4. Sketch the graph of y = x2− 2x − 8
x2− 3x − 10 including asymptotes.
Example 4.5. Determine the limit: lim
x→5−
x2− 2x − 8 x2− 3x − 10.
Example 4.6. Determine the limit: lim
x→5+
x2− 2x − 8 x2− 3x − 10.
Example 4.8. Sketch a function with the following conditions: lim
x→2+f (x) = −∞,
x→2lim−f (x) =∞
2.2 HOMEWORK
Deadline for WebAssign Homework assignments are due on the date and time given on Canvas. You are expected to bring an internet capable device and while participating in class you will submit some of the homework in class. The rest of the problems should be worked on after the section is covered and before the next class.
Keeping up with the homework is critical. The material in calculus builds on prior content and by keeping up you will find the new material easier and you will learn each topic better. By learning material well as it is covered, you will find you do not need to spend much time studying for quizzes and tests.
Deadline for Written Exercises are the beginning of the period in class on the due date on Canvas.
“Drill” problems are the type that you need to become almost automatic. “Putting it together” problems are problems use what we have learned from this section and prior sections and/or require more in-depth thought about how to apply the concepts.
Not every section will have both types of exercises.
As you work homework, your goal should be understanding why, not just how.
You should be able to explain to a classmate your solution and answer questions your classmates have about why you took the particular path you did. Of course you want your answers to be correct before your 5 tries are up on WebAssign, but your final goal should not be correctness. Do not be afraid of mistakes. Find your errors and learn why it is an error before retrying an problem.
Drill Exercises:
(3pt) 2.2 WebAssign Homework #1(2.2.006), 2(2.2.009), 3(2.2.035) (0pt) 2.2 WebAssign Homework #4(Text 2.2 p.92 # 7, 8), 5(2.2.019) (3pt) 2.2 WebAssign Homework # 6(2.2.022), 7(2.2.031), 8(2.2.040)
