Functions and Limits

  • Introduction to Functions

    In mathematics, the term Function is very famous, if we look in the historical background the term function was first used by very well known Mathematician Leibniz in 1676, who put the meaning of function in term of that dependence of one quantity on another quantity. Function is also known as the input and output […]

  • Concept of Functions

    Let A and B be any two non–empty sets. Then a function ‘’ is a rule or law which associates each element of ‘A’ to a unique element of set ‘B’. Notation: (i) Function is usually denoted by small letters i.e. etc and the Greek letter i.e. etc. (ii) If ‘’ is a function from […]

  • Examples of Functions

    Example: Find the range of the function . Solution: We have Put Thus, the domain is, Now for the range, we have For So, the range of the function is . Example: Let . Find the domain and range of . Solution: We have For For Thus, the domain is. Now for the Range, we […]

  • Types of Functions

    Constant Function: Let ‘A’ and ‘B’ be any two non–empty sets, then a function ‘’ from ‘A’ to ‘B’ is called Constant Function if and only if range of ‘’ is a singleton. Algebraic Function: The function defined by algebraic expression are called algebraic function. e.g. Polynomial Function: A function of the form Where ‘n’ […]

  • Nature of Functions

    One – One Function: Let ‘A’ and ‘B’ be any two non–empty sets then a function ‘’ from A to B is called one–one function, if and only if distinct elements of set A have distinct elements of set B. e.g. , In Mathematically, Let be a function then ‘’ is called one–one function if […]

  • Concept of Limit

    Meaning of the Phrase “Tend to Zero”: Suppose a variable ‘x’ assumes in succession a set of values. Clearly, ‘x’ is becoming smaller and smaller as n is increasing and can be made as small as we want by taking ‘n’ sufficiently large. This un-ending decrease of ‘x’ is symbolically expressed by “” and read […]

  • Definition of Limit

    Let be a real valued function if the value of the function approaches a fixed number say as approaches to a number say , we say that is the limit of function as approaches . Mathematical, it can be written in the form We read it as “limit of is as approaches to ”. If […]

  • Left Hand and Right Hand Limit

    If is a real valued function, then can approaches from two sides, the left side of and the right side of . This is illustrated with the help of diagram as shown. Left Hand Limit: If approaches from left side, i.e. from the values less than , the function is said to have left hand […]

  • Basic Theroems on Limits

    In this tutorial we give the statements of theorems on limits which will be useful in evaluating the limits. (1) The limit of a function, if exists is unique. (2) If and are any constants then, . (3) If is a constant, then for any number , . (4) For any constant and , we […]

  • Limit of a Polynomial Function

    In this tutorial we shall give idea of limit of a polynomial function of any degree which is useful to solve different polynomial functions limits. If is a polynomial function of degree , show that We have given polynomial function of degree Putting in the above equation (i), we have Taking the limit of equation […]

  • Limit of Radical Expressions

    In this tutorial we shall drive of limit of a radical expression, these types of limits usually solve by using method rationalization. Now we consider an example to evaluate Since the denominator of the given expression becomes zero at , so rationalize it before applying limits Now applying the limits, we have