Effective ways of learning & calculating limit of functions

What is a function?

A simple definition of function is that for every input, there’s an individual, possible output. In other words, every set of input has an equal set of output. One input is bound with only one output.

For example:

Suppose, we have a function ‘f’ from ‘x’ to ‘y’ then we can write the notation like

f: x🡪y

An example from real life

Engineers when designing the engine of a car, since the structure of the engine is way too complex. Engineers use simple functions to get over the complexities. Estimates and guesses are easier with the help of functions.

What is the limit of a function?

In English, limit means to ‘define a perimeter’. In mathematics, limit is defined as the boundary between an input and an output. In simple words, the start of a sequence i.e. input till the end of a sequence i.e. output. 

Limits are necessary in mathematics; calculus. They are used to delineate the concepts of integrals, derivatives and continuity. Integration calculator helps in calculating integral equations easily.

How can we determine the limit of any function?

There are different methods of finding limits

  1. Direct substitution method
  2. Factoring 
  3. Finding the LCD (lowest common denominator)
  4. Rationalizing the numerator


In this method, we get a value for a variable, we simply put it in the equation and get desired function value.

For example:

We have

x = 4 

function y = x + 10

then putting 4 in the equation

we get




It’s another method used in the limit of a function in which we find the function value by canceling out the common factors. This method helps us to make an indeterminate function into a direct evaluation function.


In LCD, we find the common factors from both the equations that are least common. Then we take an LCM and simplify the equation.

For example:

We have 

Lim x🡪0  (3/x) – (3/x2+x)

By simplifying


We get



Lim x🡪0  (3/x) – [3/(x(x+1)] ———— eq (1)

Taking least common denominator LCD



by putting it in equation eq (1)

Lim x🡪0  (3/x).x(x+1) – (3/x2+x).x(x+1)

Then we will solve the equation by simplifying it further.


It is a method in which we eliminate all the radical expressions e.g. square roots, cube roots etc. in the numerator. The key idea is to get the correct value by multiplying the original value with the suitable value. It helps us to get rid of all the radical expressions from the equation and our equation is easily solvable. 

Ways of learning limits

A limit permits us to scrutinize the propensity of a function around a given point. When the function is not defined even, we can still check the function’s tendency. 

For example
we have an equation


if we take x=1


(x2−1)/x−1 = (12-1) 1-1

       = 0/0

    = 0

Zero in the denominator indicates ‘undefined’ function.

if we take x=2


(x2−1)/x−1= (22-1)/ 2-1

    = 3/1

    = 3

Its limits exist i.e. x🡪2 we get function value ‘3’

Why do we learn limits in mathematics?

It expresses the value of a function that approaches another function. In other words, it tells us about the function’s inputs which gets nearer to some number. The concept of a limit is the foundation of calculus in mathematics. Limit calculator with steps makes it easy for you to calculate the limit of a function online.

Limits in real life:

There are several uses of limits in our daily life. It can be used to measure time with respect to infinity.

For example, if we put ice cubes in a bottle of warm water. The temperature will change. The ice will melt and the water becomes cold. And later, both the temperatures reach the room temperature.

Measuring temperatures with respect to melting point time or the time when it had attained the temperature of the room, is an example of limit in daily life.

Multiple examples can be seen while experimenting in chemistry labs when we are finding the boiling points of different chemicals or when we are guessing the dispersion of any gas with respect to another.