**Function and Limits**

Function are vital gear by way of which we describe the real world in mathematical phrases. They may be used to give an explanation for the relationship among variable quantities and hence play a crucial function within the study of calculus.

**Define Limit of a Function in Calculus**

A **Function** *f *from a set **X **to a set **Y **is a rule or a correspondence that assigns to each element **x **in **X **a completely unique element **y **in **Y**. The set **X **is known as **Domain** of *f*.

The set of corresponding elements y in ** Y **is known as the Range of

*f*. Also, except stated to the contrary, we will count on hereafter that the set

**x**and

**y**consist of actual numbers.

**What is the Real Concept of Function?**

The term Function was recognized by a **German Mathematician Leibniz** (1646-1716) to describe the dependence of one quantity on another. The subsequent examples show how this term is used:

- The area “
**A**” of a square depends on one of its sides “**x**” by the formula**A =**\boldsymbol x^{\mathbf2} , so we say that**A**is function of**x**. - The volume “
**V**” of a sphere depends on its radius “**r**” by the formula**V =**\frac{\mathbf3}{\mathbf4}\boldsymbol\pi\boldsymbol r^{\mathbf3} , So we say that**V**is a Function of**r**.

A feature is a **rule** or **correspondence** referring to two units in this sort of manner that every element inside the first set corresponds to at least one and simplest one detail inside the

- Thus in (1) above, a square of a given side has only one area.
- And in (2) above, a sphere of a given radius has only one volume.

**Notation and Value of Function**

If a variable **y** depends on a variable** x** in such a way that each of determines exactly one value f **y**, then we says that **Y** is a function.

**Note for Function **

Note: Functions are generally denoted by some letters such asf, g, h,F,G,H, and so on.

A Function can be thought as a computing machine f that takes an input **x**, operate on it in some way, and produces exactly one output **f(x)**. This output **f(x)** is called the value of f at **x** or image of **x** under **f**. The output **f(x)** is denoted by a single letter, say **y**, and we write **y = f(x)**.

**Dependent and ****Independent** **Variable in Function**

**Independent**

The Variable x is called the** independent variable** of *f*, and the variable y is called the **dependent variable** of *f*. for now onward we shall only consider the function in which the variable are real numbers and we say that f is **real valued function of real numbers**.