Profile
Settings
Refer your friends
Sign out
A limit denotes the value that a function approaches as the input reaches some value. The concept of limit is an integral part of calculus, as it is primarily used to define continuity, derivatives, and integrals.
Consider a function. f(x) = x2
When you put this in a graph, you can see that when the value of the function, f(x) = x2 approaches zero, the value of x also approaches zero.
Because there exist two ways in which l can approach a value, that is, from the right or the left, all values of x around l could take a value less than l or greater than l.
When x moves from right, we call it the right-hand limit, denoted by
Infinite Limits
Substitution method
In this method, we determine limits by substituting the value that x is approaching in the function.
Factoring method
How will you calculate the limit of rational function f as x approaches two?
When you try to solve this using the substitution method explained above,
Here, the numerator and denominator become zero, and paradoxes follow.
So, to evaluate problems of this kind, we follow a factoring method.
In the factoring method, we factor the numerator and denominator. Here, in the above problem, the denominator is factored already.
When we write the numerator in its factorial form, we arrive at the equation below,
Rationalisation or double Rationalisation method
The result indicates that problems of the above kind that involve solving the limits of rational functions require an altogether different approach and method.
Infinite limits from the right
Let f(x) be a function defined at all values in an open interval of the form (a,c).
If the value of f(x) increases without bound, as of the value of x (where x>a) approaches the number a, then we say the limit x approaches from the left is positive infinity, which is denoted as under:
These laws were created to prove that if an exponential exponent goes to infinity in the limit, the exponential function will also go to infinity. In the same way, if the exponent flows to minus infinity in the limit, the exponential will flow to zero.
Expansion in Evaluating Limits
Some essential limits are as follows:
Some of the critical expansions are as follows
Conclusion:
A limit is a value of a solution that approaches as the input approaches some value in mathematics. Limits are essential to learn before moving to core calculus and mathematical analysis. Limits are the values that are further used to define continuity, derivatives, and integrals.
Here, x→c denotes the value of x that is tending to c.
Here, the limit of the function is said to exist only when the right-hand limit is equal to the left-hand limit of the function.
