Introduction to Limits

In mathematics, a limit is defined as a value approaching the output of the given input values by a function. In calculus and mathematical analysis, the limit is important and it is used to describe integrals, derivatives, and continuity. It is used in the research process and is often concerned with the function’s behavior. Let assume, you and your friends decide to meet outside somewhere. Do you have to make all your friends live in the same place and walk the same way to this site? No not just now and then. Your all friends come to meet in one place from various parts of the town or world. Assume we have an f(x) function. The value, a function, arrives with the variable x reaching a given value, meaning that the variable an is referred to as its limit. Here, ‘a is an initial value. It’s labeled as

lim _{x ⇢ a} f(x) = 1 

The predicted function value indicated by point ‘a’ on the left is the left-hand limit of this function. It is called 

lim _{x ⇢ -a} f(x) = 1

The predicted function value indicated by point ‘a’ on the right is the right-hand limit of this function. It is called 

lim _{x ⇢ +a} f(x) = 1

Limits Definition

Let us consider a real-valued function “f” and the real number “c”, the limit is normally defined as

lim _{x ⇢ p} f(x) = 1

It is read as “the limit of f of x, as x approaches p equals L”. The “lim” shows the limit, and fact that function f(x) approaches the limit L as x approaches p is described by the right arrow.

Properties of Limits

• lim _{x ⇢ a}  k = k, where k is a constant quantity
• The value of lim _{x ⇢  a} x = a
• Value of lim _{x ⇢ a} bx + c = ba + c
• lim _{x ⇢ a} xⁿ = aⁿ if n is a positive integer.
• Value of lim _{x  ⇢ +0} 1/x^r  = +∞  
• lim _{x ⇢ −0} 1/x^r  = −∞, if r is odd
• lim _{x ⇢ −0} 1/x^r  = +∞, if r is even

Algebra of Limit

Let m and n be two functions such that their limits 

lim _{x ⇢ a} m(x) and lim _{x ⇢a} n(x) exists.

• The limit of the sum of two functions is the sum of the limits of the functions.

lim _{x  ⇢ a} [m(x) + n(x)] = lim _{x ⇢ a} m(x)  +  lim _{x ⇢ a} n(x)       

• The limit of the difference of two functions is the difference between the limits of the functions.

lim _{x  ⇢ a} [m(x) – n(x)] = lim _{x ⇢ a} m(x) – lim _{x ⇢ a} n(x)       

• The limit of product of two functions is the product of the limits of the functions.  

lim _{x  ⇢ a} [m(x) ×  n(x)] = lim _{x ⇢ a} m(x)  ×  lim _{x ⇢ a} n(x)       

• The limit of quotient of two functions is the quotient of the limits of the functions.

lim _{x  ⇢ a} [m(x) ÷ n(x)] = lim _{x ⇢ a} m(x)  ÷  lim _{x ⇢ a} n(x)      

• The limit of product of a function m(x) with a constant, n(x) = α is α times the limit of m(x).    

lim _{x ⇢ a} [α.m(x)] = α.lim _{x ⇢ a} m(x)   

Limit of Polynomial Function

Consider a polynomial function, f(x) = a₀ + a₁x + a₂x² + … + a_nxⁿ. Here, a₀, a₁, … , a_n are all constants. At any point x = a, the limit of this polynomial function is

lim _{x ⇢ a} f(x) = lim _{x ⇢ a} [a₀ + a₁x + a₂x² + . . .  + a_nxⁿ] = lim _{x ⇢ a} a₀ + a₁lim _{x ⇢ a} x + a₂lim _{x ⇢ a} x² + . . . + a_nlim _{x ⇢ a} xⁿ

or,

lim _{x ⇢ a}  a₀ + a₁x + a₂x² + . . . + a_nxⁿ = f(a)

Limit of Rational Function

The limit of any rational function of the type m(x)/n(x), where n(x) ≠ 0 and m(x) and n(x) are polynomial functions, is:

lim _{x ⇢ a} [m(x)/n(x)] = lim _{x ⇢ a} m(x)/lim _{x ⇢ a} n(x) = m(a)/m(b)          

The very first step to find the limit of a rational function is to check if it is reduced to the form 0/0 at some point. If it is so, then we need to do some adjustments so that one can calculate the value of the limit. This can be done by Canceling the factor which causes the limit to be of the form 0/0. Assume a function, f(x) = (x² + 4x + 4)/(x² − 4). Taking limit over it for x = −2, the function is of the form 0/0,

lim _{x ⇢ -2} f(x) = lim _{x ⇢ 0} [(x² + 4x + 4)]

= lim _{x ⇢ -2} [( x + 2)²/(x – 2)( x – 2)]

= lim _{x ⇢ -2} [(x – 2)/(x – 2)]

= 0/-3 ( ≠ 0/0 ) = 0

Applying the L – Hospital’s Rule

Differentiating both the numerator and the denominator of the rational function until the value of the limit is not of the form 0/0. Assume a function, f(x) = sin x/x. Taking limit over it for x = 0, the function is of the form 0/0. Taking the differentiation of both sin x and x with respect to x in the limit, lim _{x ⇢ 0} sin x/x  reduces to lim _{x ⇢ 0} cos x/1 = 1.(cos 0 = 1)

Solved Examples 

Question 1: lim _{x ⇢ 6} x/3 

Solution:

This can be easily done using the substitution method  lim _{x ⇢ 6} x/3 = 6/3 = 2

Question 2: lim _{x ⇢ 2} x² – 4/x² – 2

Solution:

x² – 4 can factorized in (x² – 2²) = ( x – 2 )( x – 2 ) 

= lim _{x ⇢ 2} x² – 4/x² – 2 

= lim _{x ⇢ 2} (x- 2)(x – 2)/x – 2 = lim _{x ⇢ 2} x + 2/1

= 4/1

= 4

Question 3: lim _{x ⇢ 1/2} 2x – 1/4x² – 1

Solution:

4x² – 1 can be factorized in a (2x²) – (1²) = (2x + 1) (2x – 1) 

Then,

= lim _{x ⇢ 1/2}  2x – 1/4x² – 1

= lim _{x ⇢ 1/2}  2x- 1/(2x – 1) (2x + 1)

= lim _{x ⇢ 1/ 2} 1/(2x + 1)

= 1/2 × (1/2) + 1

= 1/2

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