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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

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
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
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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