2. Limits and Continuity 1
1.1
A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE
1.2
THE CONCEPT OF LIMIT
1.3
COMPUTATION OF LIMITS
1.4
CONTINUITY AND ITS CONSEQUENCES
1.5
LIMITS INVOLVING INFINITY; ASYMPTOTES
1.6
FORMAL DEFINITION OF THE LIMIT
1.7
LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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3. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
1.7
LIMITS AND LOSS OF SIGNIFICANCE ERRORS
Rationale
When we use a computer (or calculator), we must always keep in mind that these devices perform most computations only approximately.
Most of the time, this will cause us no difficulty whatsoever. Occasionally, however, the results of round-off errors in a string of calculations are disastrous.
In this section, we briefly investigate these errors and learn how to recognize and avoid some of them.
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4. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
1.7
LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.1
A Limit with Unusual Graphical
and Numerical Behavior
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5. ? LIMITS AND LOSS OF SIGNIFICANCE ERRORS 1.7 7.1
A Limit with Unusual Graphical
and Numerical Behavior ?
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6. ? LIMITS AND LOSS OF SIGNIFICANCE ERRORS 1.7 7.1
A Limit with Unusual Graphical
and Numerical Behavior
The deeper we look into this limit, the more erratically the function appears to behave. We use the word appears because all of the oscillatory behavior we are seeing is an illusion, created by the finite precision of the
computer used to perform the calculations and draw the graph. ?
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7. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
Computer Representation of Real Numbers
Think of computers and calculators as storing real numbers internally in scientific notation.
For example, the number 1,234,567 would be stored as × 106.
The number preceding the power of 10 is called the mantissa and the power is called the exponent.
Thus, the mantissa here is and the exponent is 6.
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8. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
Computer Representation of Real Numbers
All computing devices have finite memory and consequently have limitations on the size mantissa and exponent that they can store. (This is called finite precision.)
Many calculators carry a 14-digit mantissa and a 3-digit exponent.
On a 14-digit computer, this would suggest that the computer retains only the first 14 digits in the decimal expansion of any given number.
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9. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.2
Computer Representation of a Rational Number
Determine how 1/3 is stored internally on a 10-digit computer and how 2/3 is stored internally on a 14-digit computer.
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10. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.2
Computer Representation of a Rational Number
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11. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.3
A Computer Subtraction of Two “Close” Numbers
Compare the exact value of
with the result obtained from a calculator or computer with a 14-digit mantissa.
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12. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.3
A Computer Subtraction of Two “Close” Numbers
If this calculation is carried
out on a computer or
calculator with a 14-digit (or smaller) mantissa, both numbers on the left-hand side are stored by the computer as 1 × 1018 and hence, the difference is calculated as 0.
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13. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.4
Another Subtraction of Two “Close” Numbers
Compare the exact value of
with the result obtained from a calculator or computer with a 14-digit mantissa.
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14. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
1.7
LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.4
Another Subtraction of Two “Close” Numbers
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15. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.4
Another Subtraction of Two “Close” Numbers
If this calculation is carried out on a calculator with a 14-digit mantissa, the first number is represented as × 1020, while the second number is represented by 1.0 × 1020, due to the finite precision and rounding.
The difference between the two values is then computed as × 1020 or 10,000,000, which is, again, a very serious error.
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16. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
Loss of Significance Errors
Examples 7.3 and 7.4 illustrate gross errors caused by the subtraction of two numbers whose significant digits are very close to one another.
This type of error is called a loss-of-significant-digits error or simply a loss-of-significance error.
These are subtle, often disastrous computational errors.
Returning now to example 7.1, we will see that it was this type of error that caused the unusual behavior noted.
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17. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.5
A Loss-of-Significance Error
Using
follow the calculation of f (5 × 104) one step at a time, as a 14-digit computer would do it.
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18. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
7.5
A Loss-of-Significance Error
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19. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
Loss of Significance Errors
In the case of the function from example 7.5, we can avoid the subtraction and hence, the loss-of-significance error by rewriting the function as follows:
where we have eliminated the subtraction.
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20. LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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LIMITS AND LOSS OF SIGNIFICANCE ERRORS
Loss of Significance Errors
From the rewritten expression,
no “unusual” behavior
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