Approximating Limits to Avoid Overflow
For most 10-digit scientific calculators, the maximum that a number can be before an overflow error occurs is 9.999999999×10^99.
The Maximum Number
What is the maximum number x can be before the calculator overflows? (For reference, I used a TI-84 Plus to calculate these numbers.)
Function: Maximum Number
e^(x^2): 15.17427129
x!: 69 (most calculators only allow nonnegative integers for the factorial function)
3^x: 209.5903274
e^x: 230.2585093
2^x: 332.1928095
x^4: slightly less than 1E25 (1 × 10^25) (like 9.99999999999E24)
x^3: 2.15443469E33 (2.15443469 × 10^33)
x^2: slightly less than 1E50 (1 × 10^50) (like 9.99999999999E49)
x^(3/2): 4.641588833E66 (4.641588833 × 10^66)
Limits to Infinity
Why is this important? This could be useful in applications, say approximating values of y(x) when x approaches infinity.
Example 1:
y = ln(x) / x^3
To find the limit x can be, look at the function presented and select the "part" that has the lowest limit. In this case, that "part" is x^3. Approach to that limit.
y(1E33): 7.59853081E-98
y(1.5E33): 2.26343032E-98
y(2E33): 9.58480691E-99
y(2.1E33): 8.28498493E-99
It would appear that ln(x)/x^3 approaches 0 as x approaches ∞.
The key is to select x values that will not cause the calculator to provide the overflow error.
Example 2:
y = e^x/x^2
e^x has a max x of 230.2585093, while x^2 has a max x of 9.99999999999E49. To prevent an overflow, we must choose x values approaching the lower max, in this case, 230.2585093.
y(229.9): 1.321975623E95
y(230): 1.459738847E95
y(230.1): 1.611859E95
y(230.2): 1.779832347E95
It would appear that e^x/x^2 approaches ∞ as x approaches ∞.
Example 3:
y = (x^3 - 1) / (x! * x^2)
Since the factorial is involved, the highest x can be is 69.
But y(69) overflows in this case! Then we have to select lower x values until we can get answers. This can be a trial and error approach.
y(68): overflow
y(67): 1.83706434E-93
y(66): 1.21246228E-91
y(65): 7.88100352E-90
Looking at the results from y(65), y(66), and y(67), it appears that (x^3 - 1) / (x! * x^2) approaches 0 as x approaches ∞.
Remember that this isn't proof, but we can get an idea how a rational function will behave. We still need the tools such as the ratio, root, and other tests.
Eddie
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