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If $\underset{x\to \infty}{\text{lim}}$ f(x) = ∞ and $\underset{x\to \infty}{\text{lim}}$ g(x) = ∞, and f(x) and g(x) have the first derivatives, f '(x) and g'(x) , respectively, then $\underset{x\to \infty}{\text{lim}}$ f(x)/g(x) = f '(x)/g'(x) .
This also holds when $\underset{x\to \infty}{\text{lim}}$ f(x) = 0 and $\underset{x\to \infty}{\text{lim}}$ g(x) = 0 , instead of $\underset{x\to \infty}{\text{lim}}$ f(x) = ∞ and $\underset{x\to \infty}{\text{lim}}$ g(x) = ∞.
For example, $\underset{x\to \infty}{\text{lim}}$ x/ex = $\underset{x\to \infty}{\text{lim}}$ 1/ex = 0, because (ex)' = ex, where e is the base for the natural logarithm.
Similarly $\underset{x\to \infty}{\text{lim}}$ ln x/x = ( 1/x )/1 = $\underset{x\to \infty}{\text{lim}}$ 1/x = 0 .
Note that this rule can be applied repeatedly as long as the conditions are satisfied.
So, for example, $\underset{x\to \infty}{\text{lim}}$ x2/ex = $\underset{x\to \infty}{\text{lim}}$ 2x/ex = $\underset{x\to \infty}{\text{lim}}$ 2/ex = 0.
Sometimes, it is very necessary to compare the order of some common used functions including the following:
1 | logn | n | nlogn | n2 | 2n | n! | nn |
Now, we can use what we've learned above about the concept of big-Oh and the calculation methods to calculate the order of these functions. The result shows that each function in the above list is big-oh of the functions following them. Figure 2 displays the graphs of these functions, using a scale for the values of the functions that doubles for each successive marking on the graph.
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1. Indicate which of the following statements are correct and which are not.
a. The range of a function is a subset of the co-domain.
b. The cardinality of the domain of a function is not less than that of its range.
c. The range of a function is the image of its domain.
d. Max {f,g} is the function that takes as its value at x the larger of f(x) and g(x).
2. Indicate which of the following statements are correct and which are not.
a. $\underset{x\to \infty}{\text{lim}}$ (n2 + 3n + 5)/(4n2 + 10n + 6) = 1/4.
b. 2n is big-theta of 3n.
c. $\underset{x\to \infty}{\text{lim}}$ (10n3 + 3n2 + 500n + 100)/(2n4 + 3n3) = $\underset{x\to \infty}{\text{lim}}$ (6n + 6)/(24n2 + 18n)
d. $\underset{x\to \infty}{\text{lim}}$ f’/g’ = $\underset{x\to \infty}{\text{lim}}$ f’’/g’’. if f’’ and g’’ exist, and $\underset{x\to \infty}{\text{lim}}$ f’ and $\underset{x\to \infty}{\text{lim}}$ g’ are both equal to infinity or 0.
3. Which f is not a function from R to R in the following equations, where R is the set of real numbers? Explain why they are not a function.
a. f(x) = 1/x
b. f(x) = y such that y 2 = x
c. f(x) = x 2 – 1
4. Find the domain and range of the following functions.
a. the function that assigns to each bit string (of various lengths) the number of zeros in it.
b. the function that assigns the number of bits left over when a bit string (of various lengths) is split into bytes (which are blocks of 8 bits)
5. Determine whether each of the following functions from Z to Z is one-to-one, where Z is the set of integers.
a. f(n) = n + 2
b. f(n) = n² + n + 1
c. f(n) = n³ - 1
6. Determine whether each of the following functions from Z to Z is onto.
a. f(n) = n + 2
b. f(n) = n² + n + 1
c. f(n) = n³ - 1
7. Determine whether each of the following functions is a bijection from R to R.
a. f(x) = 2x + 3
b. f(x) = x² + 2
8. Determine whether each of the following functions from R to R is O(x) .
a. f(x) = 10
b. f(x) = 3 x + 7
c. f(x) = x ² + x + 1
d. f(x) = 5 ln x
9. Use the definition of big-oh to show that x 4 + 5 x 3 + 3 x 2 + 4 x + 6 is O(x 4 ) .
10. Show that ( x² + 2 x + 3) / ( x + 1) is O(x) .
11. Show that 5 x 4 + x² + 1 is O(x 4 /2) and x 4 /2 is O (5 x 4 + x² + 1).
12. Show that 2n is O (3n) but that 3n is not O (2n).
13. Explain what it means for a function to be O (1).
14. Give as good (i.e. small) a big- O estimate as possible for each of the following functions.
a. ( n² + 3 n + 8)( n + 1)
b. (3log n + 5 n² )( n³ + 3 n + 2)
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