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5_Limits_of_Transcendental_Functions_v2.notebook
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Limits of Transcendental Functions
Lesson objectives Teachers' notes
Topic 1.8: Determining Limits Using the Squeeze Theorem
Topic 1.6: Determining Limits Using Algebraic Manipulation
LIM1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.
Topic 1.7: Selecting Procedures for Determining Limits
Topic 1.9: Connecting Multiple Representations of Limits
LIM1.E: Determine the limits of functions using equivalent expressions for the function or the squeeze theorem.
1.C: Identify an appropriate mathematical rule or procedure based on the classification of a given expression.
2.C: Identify a reexpression of mathematical information presented in a given representation.
Teachers' notesLesson objectives
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If you research the definition of a transcendental function you will find this explanation on Wolfram’s MathWorld:
A function which is not an algebraic function. In other words, a function which "transcends," i.e., cannot be expressed in terms of, algebra. Examples of transcendental functions include the exponential function, the trigonometric functions, and the inverse functions of both.
Lesson 5: LIMITS OF TRANSCENDENTAL FUNCTIONS
EX #1: Recall that exponential equations are written in the form . You will need to find limits of exponential
functions without the aid of a graph or calculator in this course. Do you remember the rules for transformations of exponential functions? Evaluate the limits using the graphs and look for patterns
Analyzing Limits of Exponential Functions
A.
Topic 1.9: Connecting Multiple Representations of Limits
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B.
C.
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D.
EX #2: WHAT JUST HAPPENED? Did you see that? Basically, the endbehavior of any exponential function tends toward three places.
Case #1: Case #2: Case #3:
Now, use the graphs from the previous example to complete the table below:
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EX #3: Let’s summarize a few facts related to graph transformations in order to find limits without the aid of a graph. Using the information from above, write the four conditions that can occur.
EX #4: You got this! Find the limits of each of the following exponential functions.
A. B.
C. D.
E. F.
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Analyzing Limits of Trigonometric Functions
Most of the work you have done with limits to this point have dealt with polynomial or rational functions. When confronted with trigonometric functions, you will find throughout the course, that there are several methods to use. Let’s begin with the basics in this section.
EX#5: Use direct substitution to find each limit.
A. B.
Topic 1.7: Selecting Procedures for Determining Limits
EX #6: Evaluate the limit by direct substitution.
Just like some polynomial functions where a function value is not defined, yet a limit will exist…the same will occur with trigonometric functions.
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A few thoughts to consider:
If you algebraically determine that a function is undefined at a value, does that mean the limit does not exist? Can you look at the graph? How do you analyze the limit when no calculator or graph is permitted? We need to use some trig identities and rewrite the function. Try that here:
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EX #7: You Got This! Use your new skills to evaluate each limit below by first rewriting the function and using identities.
A. B.
C.
Topic 1.6: Determining Limits Using Algebraic Manipulation
HAVE YOU DISCOVERED A PATTERN IN ALL OF THIS YET? If you ALWAYS try direct substitution first, three things will occur!
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SPECIAL TRIGONOMETRIC LIMITS
Later in the course you will learn a smooth technique known as L’Hospital’s Rule. It will be a quicker method for evaluating many functions. For now, you might like to memorize these special rules, as well.
EX #8: Use a graphing calculator and the table feature to evaluate the following trigonometric limits.
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EX#9: Some algebraic tricks with properties.
A. B.
Topic 1.8: Determining Limits Using the Squeeze Theorem
EX #10: Evaluate the limit using the squeeze theorem.
A. Explain why you cannot use the Product Limit Law.
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B. Consider the domain of the sine function −1≤ sin x ≤1 . We can conclude that
Multiply through by x2 to use the Squeeze Theorem.
EX #11: Find the limit.