# Limits

The limit of a function explains the behavior of that function at a specific point. Basically, the limit is the output value of the function for which the input value approaches closer and closer to a particular point.

If a function 'f' is defined for values of 'x' about the fixed number 'a' and if 'x' tends towards 'a', the values of f(x) get closer and closer to some specific number L. Then, $\lim x\rightarrow a f(x)=L$. Here, we receive it as "the limit of f(x) as x approaches 'a' is L.

The value of a given function f(x) approaches as 'x' gets close to some value 'a'. To indicate that 'x' is approaching 'a' we write $x \rightarrow a$. If f(x) = L, when x -> a. We write $\lim x\rightarrow a f(x)=L$. Geometrically, this means that the series of rectangles which surround 'a' and which have narrower widths become smaller in height progressively and cluster about (a, L).

### Limits of Functions

The mathematical expression of the limit of a function is described as primary function in calculus and analysis and it is consider by the behavior of that function near an exacting input. For every input of x we get an output f(x) .The function f(x) has a limit L at an input n.

Whenever x is near to n but f(x) is very near to L. In other way, as x approaches near and near to n also the f(x) becomes near to L. In addition, when f is of use to every input suitably close to n, the result is an output value specifically randomly near to L. If the values are very different, output is close to n -- the limit is held to "not exist".

### Ways to Identify the Limits of a Function

- When the limit is zero, you need to first enter the value of x.
- Now, check the factors of the numerator. Find it.
- Find the factors of denominator and simplify them all.
- Now, enter the values for x and calculate f(x).
- The value you get is the limit of the function.

### Limit Theorems

To verify limit theorem let us have two functions f(x) and g(x) in such a way that $\lim x\rightarrow a f(x)=L$, $\lim x\rightarrow a g(x)=M$

**Theorem 1: **

The sum of the limits of two given functions if equal to the limit of the sum of two functions.

$\lim x\rightarrow a [f(x)+g(x)]= \lim x\rightarrow a f(x) + \lim x\rightarrow a g(x)$= L + M

**Theorem 2: **

If we multiply any function by any constant say c then $\lim x\rightarrow acf(x) = c \lim x\rightarrow af(x) = c L$

So, we can say that the limit of a constant multiple of the function f(x) is equal to the c times the limit of that function.

**Theorem 3: **

The limits of the product of any two given function is same as the product of the limits of that given functions.

$\lim x\rightarrow a[f(x) \times g(x)]= \lim x\rightarrow af(x) \times \lim x\rightarrow ag(x) = L \times M$

**Theorem 4: **

If we have the quotient of two functions then the limit that term is same as the quotient of their limits included the condition that the limit of the denominator is not equal to zero.

$\lim x\rightarrow af(x)\times g(x)=\lim x\rightarrow af(x)\times \lim x\rightarrow ag(x) = LM, M \neq 0.$

Math >> Pre-calculus