# 5.11 Limits of algebraic functions

 Page 1 / 3

Algebraic expressions comprise of polynomials, surds and rational functions. For evaluation of limits of algebraic functions, the main strategy is to work expression such that we get a form which is not indeterminate. Generally, it helps to know “indeterminate form” of expression as it is transformed in each step of evaluation process. The moment we get a determinate form, the limit of the algebraic expression is obtained by plugging limiting value of x in the expression. The approach to transform or change expression depends on whether independent variable approaches finite values or infinity.

The point of limit determines the way we approach evaluation of limit of a function. The treatment of limits involving independent variable tending to infinity is different and as such we need to distinguish these limits from others. Thus, there are two categories of limits being evaluated :

1: Limits of algebraic function when variable tends to finite value.

2: Limits of algebraic function when variable tends to infinite

## Limits of algebraic function when variable tends to finite value

In essence, we shall be using following three techniques to determine limit of algebraic expressions when variable is approaching finite value – not infinity. These methods are :

1: Simplification or rationalization (for radical functions)

2: Using standard limit form

3: Canceling linear factors (for rational function)

We should be aware that if given function is in determinate form, then we need not process the expression and obtain limit simply by plugging limiting value of x in the expression. Some problems can be alternatively solved using either of above methods.

## Simplification or rationalization (for radical functions

We simplify or rationalize (if surds are involved) and change indeterminate form to determinate form. We need to check indeterminate forms after each simplification and should stop if expression turns determinate. In addition, we may use following results for rationalizing expressions involving surds :

$\sqrt{a}-\sqrt{b}=\frac{\left(a-b\right)}{\left(\sqrt{a}+\sqrt{b}\right)}$ ${a}^{1/3}-{b}^{1/3}=\frac{\left(a-b\right)}{\left({a}^{2/3}+{a}^{1/3}{b}^{1/3}+{b}^{2/3}\right)}$

Problem : Determine limit

$\underset{x\to 1}{\overset{}{\mathrm{lim}}}\frac{\left(\sqrt{x}-1\right)\left(2x-1\right)}{2{x}^{2}-x-1}$

Solution : Here, indeterminate form is 0/0. We simplify to change indeterminate form and find limit,

$⇒\frac{\left(\sqrt{x}-1\right)\left(2x-1\right)}{2{x}^{2}-x-1}=\frac{\left(\sqrt{x}-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x+1\right)}=\frac{\left(2x-1\right)}{\left(\sqrt{x}+1\right)\left(2x+1\right)}$

This is determinate form. Plugging “1” for x, we have :

$⇒L=\frac{1}{6}$

Problem : Determine limit

$\underset{x\to 0}{\overset{}{\mathrm{lim}}}\frac{1}{x{\left(8+x\right)}^{\frac{1}{3}}}-\frac{1}{2x}$

Solution : The indeterminate form is ∞-∞. Simplifying, we have :

$f\left(x\right)=\frac{2-{\left(8+x\right)}^{\frac{1}{3}}}{2x{\left(8+x\right)}^{\frac{1}{3}}}$

We know that :

${a}^{1/3}-{b}^{1/3}=\frac{\left(a-b\right)}{\left({a}^{2/3}+{a}^{1/3}{b}^{1/3}+{b}^{2/3}\right)}$

Using this identity :

$⇒2-{\left(8+x\right)}^{\frac{1}{3}}={8}^{\frac{1}{3}}-{\left(8+x\right)}^{\frac{1}{3}}$ $=\frac{\left(8-8-x\right)}{\left({8}^{\frac{2}{3}}+{8}^{\frac{1}{3}}{8}^{\frac{1}{3}}+{8}^{\frac{2}{3}}\right)}$

Substituting in the given expression,

$=\frac{-x}{2x{\left(8+x\right)}^{\frac{1}{3}}\left({8}^{\frac{2}{3}}+{8}^{\frac{1}{3}}{8}^{\frac{1}{3}}+{8}^{\frac{2}{3}}\right)}$ $=\frac{-1}{2{\left(8+x\right)}^{\frac{1}{3}}\left({8}^{\frac{2}{3}}+{8}^{\frac{1}{3}}{8}^{\frac{1}{3}}+{8}^{\frac{2}{3}}\right)}$

This is a determinate form. Plugging “0” for x,

$L=\frac{-1}{2X{8}^{\frac{1}{3}}\left({8}^{\frac{2}{3}}+{8}^{\frac{1}{3}}{8}^{\frac{1}{3}}+{8}^{\frac{2}{3}}\right)}=-\frac{1}{48}$

Problem : Determine limit :

$\underset{x\to 0}{\overset{}{\mathrm{lim}}}\frac{\sqrt{\left(1-{x}^{2}\right)}-\sqrt{\left(1-x\right)}}{x}$

Solution : Here, indeterminate form is 0/0. We simplify to change indeterminate form and find limit,

$\frac{\sqrt{\left(1-{x}^{2}\right)}-\sqrt{\left(1-x\right)}}{x}=\frac{\left(1-{x}^{2}-1+x\right)}{x\left\{\sqrt{\left(1-{x}^{2}\right)}+\sqrt{\left(1-x\right)}\right\}}=\frac{\left(1-x\right)}{\left\{\sqrt{\left(1-{x}^{2}\right)}+\sqrt{\left(1-x\right)}\right\}}$

This simplified form is not indeterminate. Plugging “0” for “x” :

$L=\frac{1}{2}$

## Using standard limit form

There is an important algebraic form which is used as standard form. The standard form is (n is rational number) :

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x