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Graph of y = ( x + 6 ) ( x - 2 ) x + 6 .

We say that a function is continuous if there are no values of the independent variable for which the function is undefined.

Limits

We can now introduce a new notation. For the function y = ( x + 6 ) ( x - 2 ) x + 6 , we can write: lim x - 6 ( x + 6 ) ( x - 2 ) x + 6 = - 8 . This is read: the limit of ( x + 6 ) ( x - 2 ) x + 6 as x tends to -6 is 8.

Investigation : limits

If f ( x ) = x + 1 , determine:

f(-0.1)
f(-0.05)
f(-0.04)
f(-0.03)
f(-0.02)
f(-0.01)
f(0.00)
f(0.01)
f(0.02)
f(0.03)
f(0.04)
f(0.05)
f(0.1)

What do you notice about the value of f ( x ) as x gets close to 0.

Summarise the following situation by using limit notation: As x gets close to 1, the value of the function y = x + 2 gets close to 3.

  1. This is written as: lim x 1 x + 2 = 3 in limit notation.

We can also have the situation where a function has a different value depending on whether x approaches from the left or the right. An example of this is shown in [link] .

Graph of y = 1 x .

As x 0 from the left, y = 1 x approaches - . As x 0 from the right, y = 1 x approaches + . This is written in limits notation as: lim x 0 - 1 x = - for x approaching zero from the left and lim x 0 + 1 x = for x approaching zero from the right. You can calculate the limit of many different functions using a set method.

Method:

Limits: If you are required to calculate a limit like lim x a then:

  1. Simplify the expression completely.
  2. If it is possible, cancel all common terms.
  3. Let x approach a .

Determine lim x 1 10

  1. There is nothing to simplify.

  2. There are no terms to cancel.

  3. lim x 1 10 = 10

Determine lim x 2 x

  1. There is nothing to simplify.

  2. There are no terms to cancel.

  3. lim x 2 x = 2

Determine lim x 10 x 2 - 100 x - 10

  1. The numerator can be factorised. x 2 - 100 x - 10 = ( x + 10 ) ( x - 10 ) x - 10

  2. x - 10 can be cancelled from the numerator and denominator.

    ( x + 10 ) ( x - 10 ) x - 10 = x + 10

  3. lim x 10 x 2 - 100 x - 10 = 20

Average gradient and gradient at a point

In Grade 10 you learnt about average gradients on a curve. The average gradient between any two points on a curve is given by the gradient of the straight line that passes through both points. In Grade 11 you were introduced to the idea of a gradient at a single point on a curve. We saw that this was the gradient of the tangent to the curve at the given point, but we did not learn how to determine the gradient of the tangent.

Now let us consider the problem of trying to find the gradient of a tangent t to a curve with equation y = f ( x ) at a given point P .

We know how to calculate the average gradient between two points on a curve, but we need two points. The problem now is that we only have one point, namely P . To get around the problem we first consider a secant to the curve that passes through point P and another point on the curve Q . We can now find the average gradient of the curve between points P and Q .

If the x -coordinate of P is a , then the y -coordinate is f ( a ) . Similarly, if the x -coordinate of Q is a - h , then the y -coordinate is f ( a - h ) . If we choose a as x 2 and a - h as x 1 , then: y 1 = f ( a - h ) y 2 = f ( a ) . We can now calculate the average gradient as:

y 2 - y 1 x 2 - x 1 = f ( a ) - f ( a - h ) a - ( a - h ) = f ( a ) - f ( a - h ) h

Now imagine that Q moves along the curve toward P . The secant line approaches the tangent line as its limiting position. This means that the average gradient of the secant approaches the gradient of the tangent to the curve at P . In [link] we see that as point Q approaches point P , h gets closer to 0. When h = 0 , points P and Q are equal. We can now use our knowledge of limits to write this as:

gradient at P = lim h 0 f ( a ) - f ( a - h ) h .

and we say that the gradient at point P is the limit of the average gradient as Q approaches P along the curve.

Khan academy video on calculus - 1

For the function f ( x ) = 2 x 2 - 5 x , determine the gradient of the tangent to the curveat the point x = 2 .

  1. We know that the gradient at a point x is given by: lim h 0 f ( x + h ) - f ( x ) h In our case x = 2 . It is simpler to substitute x = 2 at the end of the calculation.

  2. f ( x + h ) = 2 ( x + h ) 2 - 5 ( x + h ) = 2 ( x 2 + 2 x h + h 2 ) - 5 x - 5 h = 2 x 2 + 4 x h + 2 h 2 - 5 x - 5 h
  3. lim h 0 f ( x + h ) - f ( x ) h = 2 x 2 + 4 x h + 2 h 2 - 5 x - 5 h - ( 2 x 2 - 5 x ) h ; h 0 = lim h 0 2 x 2 + 4 x h + 2 h 2 - 5 x - 5 h - 2 x 2 + 5 x h = lim h 0 4 x h + 2 h 2 - 5 h h = lim h 0 h ( 4 x + 2 h - 5 ) h = lim h 0 4 x + 2 h - 5 = 4 x - 5
  4. 4 x - 5 = 4 ( 2 ) - 5 = 3

  5. The gradient of the tangent to the curve f ( x ) = 2 x 2 - 5 x at x = 2 is 3.

For the function f ( x ) = 5 x 2 - 4 x + 1 , determine the gradient of the tangent to curve at the point x = a .

  1. We know that the gradient at a point x is given by: lim h 0 f ( x + h ) - f ( x ) h In our case x = a . It is simpler to substitute x = a at the end of the calculation.

  2. f ( x + h ) = 5 ( x + h ) 2 - 4 ( x + h ) + 1 = 5 ( x 2 + 2 x h + h 2 ) - 4 x - 4 h + 1 = 5 x 2 + 10 x h + 5 h 2 - 4 x - 4 h + 1
  3. lim h 0 f ( x + h ) - f ( x ) h = 5 x 2 + 10 x h + 5 h 2 - 4 x - 4 h + 1 - ( 5 x 2 - 4 x + 1 ) h = lim h 0 5 x 2 + 10 x h + 5 h 2 - 4 x - 4 h + 1 - 5 x 2 + 4 x - 1 h = lim h 0 10 x h + 5 h 2 - 4 h h = lim h 0 h ( 10 x + 5 h - 4 ) h = lim h 0 10 x + 5 h - 4 = 10 x - 4
  4. 10 x - 4 = 10 a - 5

  5. The gradient of the tangent to the curve f ( x ) = 5 x 2 - 4 x + 1 at x = 1 is 10 a - 5 .

Limits

Determine the following

  1. lim x 3 x 2 - 9 x + 3
  2. lim x 3 x + 3 x 2 + 3 x
  3. lim x 2 3 x 2 - 4 x 3 - x
  4. lim x 4 x 2 - x - 12 x - 4
  5. lim x 2 3 x + 1 3 x

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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