One Sided Limits: The Right and Left hand limits

One Sided Limits

Limit of a function is given by lim (x->a) f(x) = L, that is, as x approaches ‘a’ from the right hand side of a, f(x) = L and also, as x approaches a from the left hand side of a, f(x)=L.

One Sided Limits Examples
Let us consider a function f(x) = -x^2+6 for  0<=x<2 and f(x)=x-1 for  2<= x <= 4. When we graph this function, we can see that the function is discontinuous at x=2, and the two pieces of the function approach different values, we get, lim (x->2)f(x) is undefined. We can say that the limit of f(x) as x approaches 2 from the left is 2, and the limit of f(x) as x approaches from the right is 1. We can write it as, lim (x->2-) f(x) = 2 and lim (x->2+) f(x) = 1. The plus sign indicates ‘from the right side’ and the minus sign indicates ‘from the left side’.

There are two one sided limits, Right and left hand limits. Right hand limit which is defined as lim (x->a+) f(x) = L , if f(x) is as close to L as we want for all x sufficiently closer to a and x is greater than a, without actually letting x equal to a. Left Hand Limit is defined as lim(x->a-)=L, if f(x) is as close to L as we want for all x sufficiently closer to a and x is less than a, without actually letting x equal to a.

Finding One Sided Limits Algebraically
Let us consider a piecewise function f(x)= x+2 for x less than or equal to 6 and f(x) = x^2-1 for x greater than 6. The steps involved in finding one sided limits of the given piecewise function are as follows:

Left Handed limit of f(x) is given by lim (x->6-)f(x), all the values of x on the number line lie to the left of 6 which means  x is less than 6 and hence we consider the function f(x) = x+2. We plug in the value x=6 in the function which gives, 6+2 = 8

Right Handed Limit of f(x) is given by lim(x->6+) f(x),  all the x values considered on the number line lie to the right of 6, so, x is greater than 6 and hence we consider the function f(x) = x^2-1. We shall plug in the value x=6 in the function which gives, (6)^2-1 = 36-1=35
Lim(x->6)f(x) does not exist as the right handed limit is not equal to the left handed limit

Continuous Variable

Variable is just like a box where we can store the property of an object. The property of an object may be a value or a characteristic, variable takes more than one value. For example, “marks” of a student in an exam is a variable. Eye color of individuals is a variable which is measured on nominal scale. Height of individuals is also a variable which is measured on ratio scale. Variables are classified into two categories those are Continuous Variable and Discrete Variable.

Continuous Variable Definition: A variable is said to be continuous variable if takes each and every value in between two values (or limits). Continuous variable Example is given below.

• “Rain fall in a day” is a continuous variable it can take any value in between 0 and infinity.
• “Time to complete this answer” is a continuous variable it can take any value between 0 and infinity.
• “Water level in a tank” is a continuous variable it can take any value between 0 and maximum capacity of tank.

Simultaneous Equations

Equations
Any expression F(X) when equated to an expression or a constant is called an equation.
When the highest power of X is one it is called a simultaneous equation.

What are Simultaneous Equations
A set of equations having many variables, is called simultaneous equations. To solve a simultaneous equation of two variables we need minimum two equations, which are different from each other.

Simultaneous Equation

Ex: 2x+3y = 5 and 4x +3y = 10.

Solving Simultaneous Equations:

Substitution method
Algorithm:
Step 1: Express one of the variables in terms of the other.
Step 2: Substitute the value of the variable in the second equation to obtain an equation in terms of one single variable.
Step 3: Solve for the variable.
Step 4: Substitute the value of the variable in the expression to get the solution of the other variable.

Solved Example:
Solve 2x +3y =5; 5x +2y=7
Step 1: 2x= 5-3y; x= (5-3y)/2
Step 2: Substituting the value of x in the second equation we get
 5((5-3y)/2) + 2y=7 Solving for y we get
 (25-15y+4y)/2=7
 (25-11y)/2=7
 25-11y=14
 25-14=11y
 11=11y
 y=1
Step 3: Substituting the value of x in step 1 we get x=(5-3(1))/2 =1
Hence the solution set is X=1, y=1 or {1, 1}

Method 2:
Elimination Method:
Algorithm:
Step 1: Make the coefficients of one of the variables same.
Step 2: If the signs are same, subtract and if the signs are different add them to obtain a linear equation of one variable and solve for it.
Step 3: Substitute the obtained value in any one of the equations to solve for the other variable.
Solved Example:
4x+ 3y = 11; 3x +2y=8
4x+3y=11  --------> multiply 2
3x+2y=8  --------> multiply 3
8x + 6y =22
9x + 6y =24 (as the signs are same subt)
(-)   (-)     (-)
-x = -2 or x=2
Substituting the value of x in the first equation we get
4(2) +3y =11;
8+3y=11;
3y= 11-8;
3y=3; y=1.
The solution set is x= 2 and y = 1.
Cross Multiplication Method
Algorithm:
Step 1: Rearrange the equations as shown.
If the equations are as follows

Step 2: Cross-multiply and subtract starting from top end corner of the denominator and write as follows

Step 3: 
Equate the x term with constant term and solve for x. Similarly equate y term with constant term and solve for y.
Solved Example:
2x +3y = 7
6x -5y =11

Cross-multiplying and subtracting we get

Product Rule for Derivatives

Product Rule for Derivatives:

The Product Rule helps us to differentiate a function when the function is a product of two distinct functions.

 Product Rule is necessary whenever we need to differentiate a function which is a product of two sub-functions that are easier to differentiate.

Let us consider some examples for a better understanding of the Product Rule for Derivatives

Associative Property

Definition of associative property:

Associative property pertains to number, sets and many other mathematical components. The word associative is derived from the word ‘association’, which means being related to. In math, we define associative property for various operations on numbers or sets etc.

Associative Property

What is associative propery?

All types of numbers, whether natural, whole, rational, irrational and real numbers have associative properties. There are mainly two types of associative property for numbers. They are:

1. Associative property of addition:
Addition associative property definition is as follows: Consider three numbers a, b and c. then, (a+b) + c = a + (b+c). Meaning we associate a with b first then associate the sum with c or we associate a with sum of b and c, either ways, the result would be the same.
For example: (1/2 + 1/3) + ¼ = ½ + (1/3 + ¼) = 13/12

2. Associative property of multiplication:
Numbers obey associative property in multiplication as well. Consider three numbers, p,q and r, then (p x q) x r = p x (q x r). That means whether we take product of p and q and then multiply with r or we multiply p with product of q and r, either ways we would get the same answer.

For example: (1/2 x 1/3) x ¼ = ½ x (1/3 x ¼) = 1/24
Associative property with respect to sets:

Consider three non disjoint sets, A, B and C. Then both the operations of union and intersection on these sets is associative. That means, A U (B U C) = (A U B) U C and
A ∩ (B ∩ C) = (A ∩ B) ∩ C.

Example: Consider the following sets: A = {1,2,3,4,5}, B = {2,4,6,8,10}, C = {3,4,5,6,7}
Then, A U (B U C) = {1,2,3,4,5} U ({2,4,6,8,10} U {3,4,5,6,7}) = {1,2,3,4,5,6,7,8,10}
And (A U B) U C = ({1,2,3,4,5} U {2,4,6,8,10}) U {3,4,5,6,7} = {1,2,3,4,5,6,7,8,10}
So we see that both are equal. Similarly .
A ∩ (B ∩ C) = {1,2,3,4,5} ∩  ({2,4,6,8,10} ∩  {3,4,5,6,7}) = {4,6} and
(A ∩ B) ∩ C = ({1,2,3,4,5} ∩  {2,4,6,8,10}) ∩  {3,4,5,6,7} = {4,6}

Algebra Order of Operations

Let's learn about the Algebra Order of Operations in today's post.

We all know that algebra is a study of expressions and variables and its relative operations. The algebra order of operations is also stated as bodmas, pemdas or bedmas. These are nothing but the different names for a single rule used in algebra. Let's elaborate one:

PEMDAS is the order of operations which can be explained as:
P : Parentheses
E : Exponents
M: Multiply
D : Divide
A : Add
S : Subtract

This rule is pretty much connected with algebraic symbols and terms. For more help one can connect with an online tutor anytime and get your required help. Not just in algebra but in other topics like pre calculus tutoring and so on as well.

Do post your comments.

Proper and Improper Fractions

Let's understand the two main types of fractions here: Proper fraction and improper fraction. Fraction is a rational number having numerator and denominator and based on these two, proper and improper fractions are classified.

Proper fractions are those fractions where numerator is smaller than a denominator. As for example: 2/3, 4/5 and so on.

Now let's understand what is an improper fraction. Improper fractions are those where the denominators are greater than the numerator. As for example: 5/2, 7/6 and so on.

For more help with this, get help from an online tutor. You can avail to fractions help and online geometry tutoring and others as well.

Next time we will learn about some other concepts. Till then enjoy working out proper and improper fractions. Do post your comments.