Saturday, June 27, 2015

Graphs of Logarithmic Functions


1. Graph of a function f(x) = y = logax (a>1) and f(x) = y = logax (0<a<1)

2. Graph of a function f(x) = y = log2x and f(x) = y = log3x
 


3. Graph of a function f(x) = y = log0.5x

4. Graph of a function f(x) = y = logx
5. Graph of a function (i) f(x) = y = log2x
(ii) f(x) = y = logex
(iii) f(x) = y = log3x


Neighbourhood of a Point

Definition: Let aR. If δ > 0 then the open interval (a −δ, a +δ) is called the neighbourhood (δ - nbd) of the point a. It is denoted by
Nδ (a) a is called the centre and δ is called the radius of the neighbourhood.



The set Nδ (a) − {a} is called a deleted δ -neighbourhood of the point a.
Nδ (a) − {a} = (a − δ, a) (a, a + δ) = {xR: 0 <| x − a | < δ} 

Note: (a − δ, a) is called left δ -neighbourhood, (a, a + δ) is called right δ - neighbourhood of a.

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Friday, June 26, 2015

Limits – Intervals – Definition

Definition:
Let a, bR and a < b. Then the set {xR: a≤ x≤ b} is called a closed interval. 
It is denoted by [a, b]. Thus closed interval [a, b] = { xR: a ≤ x ≤ b}. It is geometrically represented by 



  
Open interval (a,b) = {x∈R: a < x < b}. It is geometrically represented by



   
Left open interval (a, b] = {x∈R: a < x ≤ b}. It is geometrically represented by


    

Right open interval [a, b) = {x∈R: a ≤ x < b}. It is geometrically represented by



 
[a,∞) = {x∈R : x ≥ a} = {x∈ R : a ≤ x < ∞} It is geometrically represented by




(a, ∞) = {x∈R : x > a} = {x∈R : a < x < ∞}




(−∞, a] = {x∈R : x ≤ a} = {x∈ R : −∞ < x < a}



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