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and read it as, The summation of the terms ai, from i = 1 to n If we have an infinite series a1 + a2 + a3 + a4 + a5 + then we can symbolize it by writing
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and read it as, The summation of the terms ai, from i = 1 to infinity
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Question 19-9
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What is the limit of an infinite sequence, an infinite series, a relation, or a function How can we symbolize the fact that the limit of x 5, as a real number x approaches infinity, is equal to 0 What does the term convergent mean in relation to an infinite sequence or series
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A limit is a value that an infinite sequence, an infinite series, a relation, or a function approaches, but does not necessarily reach We can symbolize the fact that the limit of x 5, as a real-number variable x approaches infinity, is equal to 0 by writing
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Lim x 5 = 0
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An infinite sequence is convergent if and only if, as we move along the sequence from term to term, the values of the terms approach a definable limit An infinite series is convergent if and
Part Two 435
only if, as we move along the series from term to term, the values of the partial sums approach a definable limit
Question 19-10
What is the right-hand limit of a function at a point What is the left-hand limit of a function at a point
Answer 19-10
The right-hand limit of a function at a point is the value that the function approaches as we move toward the point from the positive direction We denote a right-hand limit by writing a small plus sign at the end of the subscript For example, if we approach the point where x = 0 along the x axis from the positive side, then the expression
Lim ln x
x 0+
refers to the limit of the natural logarithm of x, as x approaches 0 from the right (You might immediately see that this particular limit is not defined) The left-hand limit of a function at a point is the value that the function approaches as we move toward the point from the negative direction We denote a left-hand limit by writing a small minus sign at the end of the subscript For example, if we approach the point where x = 3 along the x axis from the negative side, then the expression
Lim ex
x 3
refers to the limit of the natural exponential of x, as x approaches 3 from the left (This particular limit happens to be defined, and is equal to e3)
Final Exam
This exam is designed to test your general knowledge, not to measure how fast you can perform calculations A good score is at least 80 correct answers The answers are listed in App C This test is long, so don t try to take it in a single session Feel free to draw diagrams, sketch graphs, or use a calculator But don t look back at the text or refer to outside information sources 1 Under what conditions is the dot product of two nonzero polar-plane vectors equal to 0 (a) When the two vectors point in the same direction (b) When the two vectors point in opposite directions (c) When the two vectors have equal magnitude (d) When the two vectors are mutually perpendicular (e) Under more than one of the above conditions (a), (b), (c), and (d) 2 The cosecant function is singular when the input, in radians, is equal to (a) 0 (b) p /6 (c) p /4 (d) p /3 (e) p /2 3 The conjugate of 8 j 6 is (a) undefined (b) 8 + j 6 (c) 6 j 8 (d) 6 + j 8 (e) the pure real number 10
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