# Introduction, limits, average gradient  (Page 3/3)

 Page 3 / 3 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 $y=\frac{\left(x+6\right)\left(x-2\right)}{x+6}$ , we can write: ${lim}_{x\to -6}\frac{\left(x+6\right)\left(x-2\right)}{x+6}=-8.$ This is read: the limit of $\frac{\left(x+6\right)\left(x-2\right)}{x+6}$ as $x$ tends to -6 is 8.

## Investigation : limits

If $f\left(x\right)=x+1$ , 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 $f\left(x\right)$ 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: ${lim}_{x\to 1}x+2=3$ 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 $x\to 0$ from the left, $y=\frac{1}{x}$ approaches $-\infty$ . As $x\to 0$ from the right, $y=\frac{1}{x}$ approaches $+\infty$ . This is written in limits notation as: ${lim}_{x\to {0}^{-}}\frac{1}{x}=-\infty$ for $x$ approaching zero from the left and ${lim}_{x\to {0}^{+}}\frac{1}{x}=\infty$ 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 ${lim}_{x\to a}$ then:

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

Determine ${lim}_{x\to 1}10$

1. There is nothing to simplify.

2. There are no terms to cancel.

3. ${lim}_{x\to 1}10=10$

Determine ${lim}_{x\to 2}x$

1. There is nothing to simplify.

2. There are no terms to cancel.

3. ${lim}_{x\to 2}x=2$

Determine ${lim}_{x\to 10}\frac{{x}^{2}-100}{x-10}$

1. The numerator can be factorised. $\frac{{x}^{2}-100}{x-10}=\frac{\left(x+10\right)\left(x-10\right)}{x-10}$

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

$\frac{\left(x+10\right)\left(x-10\right)}{x-10}=x+10$

3. ${lim}_{x\to 10}\frac{{x}^{2}-100}{x-10}=20$

## 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 $y=f\left(x\right)$ 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 $f\left(a\right)$ . Similarly, if the $x$ -coordinate of $Q$ is $a-h$ , then the $y$ -coordinate is $f\left(a-h\right)$ . If we choose $a$ as ${x}_{2}$ and $a-h$ as ${x}_{1}$ , then: ${y}_{1}=f\left(a-h\right)$ ${y}_{2}=f\left(a\right).$ We can now calculate the average gradient as:

$\begin{array}{ccc}\hfill \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}& =& \frac{f\left(a\right)-f\left(a-h\right)}{a-\left(a-h\right)}\hfill \\ & =& \frac{f\left(a\right)-f\left(a-h\right)}{h}\hfill \end{array}$

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:

$\mathrm{gradient at P}=\underset{\mathrm{h}\to 0}{lim}\frac{\mathrm{f}\left(\mathrm{a}\right)-\mathrm{f}\left(\mathrm{a}-\mathrm{h}\right)}{\mathrm{h}}.$

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

For the function $f\left(x\right)=2{x}^{2}-5x$ , 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: ${lim}_{h\to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$ In our case $x=2$ . It is simpler to substitute $x=2$ at the end of the calculation.

2. $\begin{array}{ccc}\hfill f\left(x+h\right)& =& 2{\left(x+h\right)}^{2}-5\left(x+h\right)\hfill \\ & =& 2\left({x}^{2}+2xh+{h}^{2}\right)-5x-5h\hfill \\ & =& 2{x}^{2}+4xh+2{h}^{2}-5x-5h\hfill \end{array}$
3. $\begin{array}{ccc}\hfill \underset{h\to 0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}& =& \frac{2{x}^{2}+4xh+2{h}^{2}-5x-5h-\left(2{x}^{2}-5x\right)}{h};\phantom{\rule{1.em}{0ex}}h\ne 0\hfill \\ & =& \underset{h\to 0}{lim}\frac{2{x}^{2}+4xh+2{h}^{2}-5x-5h-2{x}^{2}+5x}{h}\hfill \\ & =& \underset{h\to 0}{lim}\frac{4xh+2{h}^{2}-5h}{h}\hfill \\ & =& \underset{h\to 0}{lim}\frac{h\left(4x+2h-5\right)}{h}\hfill \\ & =& \underset{h\to 0}{lim}4x+2h-5\hfill \\ & =& 4x-5\hfill \end{array}$
4. $4x-5=4\left(2\right)-5=3$

5. The gradient of the tangent to the curve $f\left(x\right)=2{x}^{2}-5x$ at $x=2$ is 3.

For the function $f\left(x\right)=5{x}^{2}-4x+1$ , 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: ${lim}_{h\to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$ In our case $x=a$ . It is simpler to substitute $x=a$ at the end of the calculation.

2. $\begin{array}{ccc}\hfill f\left(x+h\right)& =& 5{\left(x+h\right)}^{2}-4\left(x+h\right)+1\hfill \\ & =& 5\left({x}^{2}+2xh+{h}^{2}\right)-4x-4h+1\hfill \\ & =& 5{x}^{2}+10xh+5{h}^{2}-4x-4h+1\hfill \end{array}$
3. $\begin{array}{ccc}\hfill \underset{h\to 0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}& =& \frac{5{x}^{2}+10xh+5{h}^{2}-4x-4h+1-\left(5{x}^{2}-4x+1\right)}{h}\hfill \\ & =& \underset{h\to 0}{lim}\frac{5{x}^{2}+10xh+5{h}^{2}-4x-4h+1-5{x}^{2}+4x-1}{h}\hfill \\ & =& \underset{h\to 0}{lim}\frac{10xh+5{h}^{2}-4h}{h}\hfill \\ & =& \underset{h\to 0}{lim}\frac{h\left(10x+5h-4\right)}{h}\hfill \\ & =& \underset{h\to 0}{lim}10x+5h-4\hfill \\ & =& 10x-4\hfill \end{array}$
4. $10x-4=10a-5$

5. The gradient of the tangent to the curve $f\left(x\right)=5{x}^{2}-4x+1$ at $x=1$ is $10a-5$ .

## Limits

Determine the following

1. ${lim}_{x\to 3}\frac{{x}^{2}-9}{x+3}$
2. ${lim}_{x\to 3}\frac{x+3}{{x}^{2}+3x}$
3. ${lim}_{x\to 2}\frac{3{x}^{2}-4x}{3-x}$
4. ${lim}_{x\to 4}\frac{{x}^{2}-x-12}{x-4}$
5. ${lim}_{x\to 2}3x+\frac{1}{3x}$

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
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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?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
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