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Theorem 5 ( l'hospital ):

 If  lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} f(x)  = ∞ and lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} g(x)  = ∞, and  f(x)  and  g(x)  have the first derivatives,   f '(x)   and   g'(x) ,  respectively,  then lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} f(x)/g(x)  = f '(x)/g'(x) .

This also holds when lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} f(x)  = 0 and lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} g(x)  = 0 ,   instead of lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} f(x)  = ∞ and lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} g(x)  = ∞.

For example, lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} x/ex   = lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} 1/ex   = 0,   because (ex)' = ex,   where e is the base for the natural logarithm.

Similarly lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} ln x/x   = ( 1/x )/1   = lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} 1/x   = 0 .

Note that this rule can be applied repeatedly as long as the conditions are satisfied.

So, for example, lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} x2/ex = lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} 2x/ex = lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} 2/ex = 0.

Summary of big – oh

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.

***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.***

Questions and exercises

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. lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} (n2 + 3n + 5)/(4n2 + 10n + 6) = 1/4.

b. 2n is big-theta of 3n.

c. lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} (10n3 + 3n2 + 500n + 100)/(2n4 + 3n3) = lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} (6n + 6)/(24n2 + 18n)

d. lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} f’/g’ = lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} f’’/g’’. if f’’ and g’’ exist, and lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} f’ and lim x size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } } {} 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 ( + 2 x + 3) / ( x + 1) is O(x) .

11. Show that 5 x 4 + + 1 is  O(x 4 /2) and x 4 /2 is  O (5 x 4 + + 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. ( + 3 n + 8)( n + 1)

b. (3log n + 5 )( + 3 n + 2)

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
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