Learning:
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A Cloud Chamber of the Mind

March 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

5:20 am Ballina NSW Australia

This is a continuation of my notes for "Calculus".

Continuation of Exercises For Section 1.3 [p. 64 - 66]

63. . This limit can be evaluated as the product of two terms: , and . Both terms have a limit of 0. Therefore their product will also have a limit of 0. Mathematica agrees.

64. . This limit has the form of p/(-1) or -p when using direct substitution. Mathematica agrees.

65. . This limit may be reduced to the limit for sin x, which is 1. Mathematica agrees.

66. . This limit may be reduced to , which is negative the square root of 2. Mathematica agrees.

67. This limit has the form 1 times 1 = 1. Mathematica agrees.

The next exercise that looks interesting is #85.

85 and 86. (View Mathematica files).

92.

Exercises 93 - 96 involve proofs. I will let this go for the moment.

97. If f(x) < g(x) for all x not equal to a, is it true that limit of f < limit of g as x approaches a? Answer is no. It is possible that the two limits could be equal.

Exercises 98 - 104 involve proofs. I will let this go.

105. Consider the function .

(a) Find the domain of f. f is clearly undefined for x = 0. Since the secant is equal to 1/cos x, the function is also undefined whenever the cosine is equal to 0 (i.e. whenever c is a multiple of p/2)

(b) Looking at the graph for this function that Mathematica draws, the domain appears to be valid for all x since there are no error messages. On the other hand the graph clearly approaches plus and minus infinity at many points.

(c) The limit of f as x approaches 0 appears to be 1/2.

(d) BUT I am not able to show this analytically!

Here are the results using Mathematica (Web, Mathematica). 7:40 am

Reflection: This was a great session. Difficulties: I am not able to show analytically that the limit of (sec x - 1)/x^2 is 1/2.