Introduction to Limits

The concept of Limits defines continuity, derivatives, and integrals which forms the cornerstone of calculus. Limits describe, evaluate and reveal how a function behaves as it approaches a point, instead of evaluating it at just that particular point. 

Let’s take an example and understand the concept better.  

f(x) = x2- 1x-1

When x is 1,

f(x) = 12- 11-1

f(x) = 00

The function becomes indeterminate at x=1. Instead of evaluating the function when x=1, let’s evaluate the function as x approaches 1 at 0.25, 0.5, 0.9, etc. 

Now, as x approaches 1, the value of the function approaches 2, though it becomes indeterminate at x=1. 

Mathematically, this is denoted as:

x1x2-1x-1=2

This means that irrespective of what happens when the function approaches an exact point, say x=1, it gets closer to 2. 

On the other side, when x approaches 1 from 2, the limit of value of the function gets closer to 2. 

Thus, the value that a function attains as a variable, say x approaches a particular value, is called the limits of the function. If a is the limit of the function, we say it as xaf(X)=L.

In other words, if a function f(x) is defined on an interval around a, the limit of a function as ‘x’ approaches ‘a’ is ‘L’. ‘x’ is close to ‘a’ but not equal to ‘a’. We define this as 

xaf(X)=L.

  Right-hand and Left-hand Limits. 

The expected value of the function to the left of value ‘a’ is the left-hand limit. We can say that the value of limit of function f is  close to L1 as x approaches a, and x

 xa-f(X)=L1

The expected value of the function to the right of value ‘a’ is the right-hand limit. We can say that the value of limit of function f is  close to L2as x approaches a, and x>a. This is denoted as

 xa+f(X)=L2.

Existence of a limit

It is important to note that only when the value of the function approaches a finite quantity from the left hand side and right hand side at a point, the limits of the function exist at that point.  

You must now be able to answer

1. What is the concept of limit and its role in evaluating the value of functions?

2. How do we determine the value of the limit of a function at a particular point?

3. Define right-hand and left-hand limits for a function at x=a?

4. What is the meaning of xaf(X)=P?

Properties of Limits

1. x0-1xn = -,  if n is odd

2. x0+1xn = +,  if n is even

3. Given that P is a constant, xa+P=P

4. Value of xa+x  is a. 

5. Value of xa+px+q = pa+q

6. xa+xn = an, if n is a positive integer. 

7. If p is a constant, then  xa+(pf(x)) =p(xa+f(x))

Algebra of Limits

If f(x) and g(x) are two functions and the limit of f(x) and g(x) exists, then the limit of the sum of two functions is equal to the sum of limits of the functions. 

8. xa(f(x) + g(x)) =xaf(x) + xag(x)

If f(x) and g(x) are two functions and the limit of f(x) and g(x) exists, then the limit of the product of two functions is equal to the product of limits of the functions.

9. xa(f(x) . g(x)) =xaf(x) . xag(x)

If f(x) and g(x) are two functions and the limit of f(x) and g(x) exists, then the limit of the difference of two functions is equal to the difference of limits of the functions.

10. xa(f(x) – g(x)) =xaf(x) – xag(x)

If f(x) and g(x) are two functions and the limit of f(x) and g(x) exists, then the limit of the quotient of two functions is equal to the quotient of limits of the functions.

11. xa(f(x)g(x)) =xaf(x)xag(x) providedxag(x) 

Limits of a Polynomial Function

While finding the value of limits of a function, you may come across functions with polynomials. Polynomials have two or more terms with a constant and a variable raised to different powers.

A polynomial function is represented as f(x) = anxn +  an-1xn-1  + … +  a2x2  +  a1x  + a0, where an, an-1, ..a0 are constants. 

Example – f(x) = x3+3×2+6

How to calculate the limits of a polynomial function?

1. Break the polynomial into various terms using the above discussed algebraic properties of limits. 

2. Find the limits of the broken-up, individual terms

3. Sum the limits in step 2

Limits of a Rational Function

A rational function is a function that can be expressed as the ratio of two polynomial functions, where the denominator polynomial is not equal to zero. 

The function f(x) = p(x)q(x)is a rational function, where q(x) is not equal to zero. 

Example f(x) = x3+3×2+6×2-4, and the domain of this function is all values of x except for x=2 and x=-2.

How to calculate the limits of a rational function?

1. Use substitution method – directly substitute the value of the variable in the interval to determine the value of limits for that function

2. When the value of limits of the function as calculated by substitution method as in step 1 becomes indeterminate, due to the both numerator and denominator becoming zero, use factorisation method 

   1. Factorize the numerator, 

   2. Cancel common terms in numerator and denominator

   3. Substitute the value for the variable. 

3. When both the substitution and the factoring methods fail, use the conjugate method. Multiply both the numerator and denominator by the conjugate of the radical function. 

4. If you are not able to determine the value of limits of a function using the above three methods, then use L’Hopital’s rule.

   Conclusion

   To master Calculus, you need to understand the concepts of limits, continuity and derivatives. With limits, we can find the slope of two same points, add up infinite numbers, and the strength of electric or gravitational fields.