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SECTION 2.2Finding Limits Graphically & Numerically
What’s the point of Calculus?
What’s the point of Calculus?
What’s the point of Calculus?
What’s the point of Calculus?
The Concept of a “Limit”
lim𝑥→𝑐
𝑓 (𝑥 )=L
Example 1Use a table of values to estimate the limit. Confirm graphically.
-5.1 -5.01 -5.001 -5 -4.999 -4.99 -4.9
Example 2(#18-26 even in your book)
18.
20.
Example 2 (cont.)(#18-26 even in your book)
22.
24.
Example 2 (cont.)(#18-26 even in your book)
26.
Example 3(#28 in your book)
a.
b.
c.
d.
e.
f.
g.
h.
(#30 in your book)
Use the graph of to identify the values of for which exists.
Example 4
Example 5Sketch a graph of a function with the given properties.
DNE
SECTION 2.3Evaluating Limits Analytically
Theorems Involving Limits
Theorem 2.1 Some Basic Limits (p. 79)
Let and be real numbers and let be a positive integer.
1. 2. 3.
Theorems Involving Limits
Theorem 2.2 Properties of Limits (p. 79)
Let and be real numbers, let be a positive integer, and let and be functions.
1. Scalar Multiple:
2. Sum or difference:
3. Product:
4. Quotient: , provided
5. Power:
Theorems Involving Limits (cont.)
Theorem 2.3 Limits of Polynomial and Rational Functions (p. 80)
If is a polynomial and is a real number, then
.
If is a rational function given by and is a real number such that , then
.
Theorems Involving LimitsTheorem 2.4 The Limit of a Function Involving a Radical(p. 80)
Let be a positive integer. The following limit is valid for all if is odd, and is valid for if is even.
Theorem 2.5 The Limit of a Composite Function (p. 81)
Example 1Find .
Example 2Find .
Example 3Find .
Example 4Find .
Other Theorems Involving Limits
• Theorem 2.6 deals with finding the limits of trigonometric, exponential, and logarithmic functions.
• Theorem 2.7 talks about fnc.’s that agree at all but one point.
• Theorem 2.8 is the Squeeze Theorem.
Example 5Find .
Theorem 2.9
1. 2. 3. Find
Example 5
Find given .
Limits of Transcendental Functions
Example 6Find the limit if it exists.
a.
b.
Functions Agreeing at All But One Point
Example 7Find .