INTRODUCTION TO LIMITS

BRIEF HISTORY CALCULUS

A system of mathematics analysis developed independently by Leibnitz and Sir Isaac Newton.

Leibnitz was first to publish his theory in 1684, some years after Newton had, in 1666, made known his theory of Fluxions. Newton’s work was not printed until 1693.

The word calculus is borne out of the phrase rate of change. Differential calculus deals with  to rate of change of a variable function. When treating the theory and application of integral, particularly their evaluation and derivation, it is called integral calculus

The symbols Newton used for differentiation were not those which we use today, but have been replaced by those suggested by Gottfried, Wilhem Leibnitz (1646  1716), the German mathematician and Newton Contemporary.

 

LIMITS

Definition : The limit of a function for a given value of the variable  is that constant to which the function continually approaches as the variable approaches the given the value such that the difference between the constant and the function may be made small as we please by making the variable approach sufficiently near to its assigned value

OR

If a function f(x) gets closer and closer to a number L as x gets closers to c from both sides, then L is the limit  of L(x) as x approaches c. The behaviour is expressed by writing

  

RIGHT HANDED AND LEFT HANDED LIMITS

There was no restriction made as to how x should be c. It is sometimes necessary to restrict the approach. Considering x and c as point on the real axis where c is fixed and x is moving, then x can approach c from right or left. We indicate this respective approach by writing

                         

,we call L₁ and L₂  respectively the right and left hand limits of f(x) at a c and denote them  f(a₊) and f(a)

In general,

A function f(x) is said to possess a right hand limit as x approaches c from values higher than c  and is expressed as  

A function f(x) is said to possess a left hand limit as x approaches c from values lower than c is expressed as  

NOTE: A function will have a limiting value only if its right hand limit equals its left hand limit. That is, we have  

ALGEBRAIC PROPERTIES OF LIMITS

Limits obey certain algebraic rules that can be used in computations 

 

 

EVALUATION OF LIMITS

Limits can be found directly or by formula. In most cases, when the given given function is a quotient ,one can find it difficult to find the limit f te function directly because it may give unreasonable solution like  

 

1.  Direct Substitution: Directly substitute the value into the expression and if we get a finite

number  then the finite number is the limit of the given expression.

Example1

Evaluate the following limits

 

2. Factorisation

EXAMPLE 2

Evaluate the following limits

 

3. Method of Rationalisation: In functions which involve square roots, rationalization of either

    numerator or denominator will simplify the computation of the limits

 

EXAMPLE 3

Evaluate the following limits

 

 

INFINITE LIMIT OF A FUNCTION

Let f(x) be a function of x. If the values of the function f(x) approach the number L as x increases without bound, we write

                         

Similarly, if the value of f(x) approach the number M as x decrease without bound, we write

                         

 

RECIPROCAL POWER RULES

For constant A and k, with k > 0

                         

Procedure For Evaluating A Limit At Infinity Of  

Step1. Divide each term in f(x) by the highest power of x^k that appear  in the denominator polynomial h(x)

Step 2. Compute  or  using algebraic properties of limit and reciprocal power rule.

EXAMPLE 4

 

TRIGONOMETRIC LIMITS

The following limit should be taken note of when evaluating limiting values of a trigonometric function

 

EXAMPLE4

Evaluate the following limits:

 

 

 

 

EXAMPLE 5

Evaluate the following limits

 

CONTINUITY

A singlevalued function f(x) is said to be continuous at a value a of its domain provided

1                     f(a) is defined

2                      exist

3                      

If f(x) is continuous at each point of an interval ,it is said to continuous on the interval . A function that is not continuous at a point is to be discontinuous at that point.

If f(x) and g(x) are two function that are continuous x = a, the following function are continuous at  x = a