**Introduction to Limit Calculus: A Brief Guide With Examples**

In calculus and mathematical analysis, limits
play a very essential role in defining the Taylor series integrals and
differentiation continuity. Limit is a well-known term for calculus and is very
helpful in finding the function at a particular point. And also check the
behavior of the function.

Limit is very useful in real life, like measuring
the strength of the electric and gravitational fields. In this section, we will discuss the
definition of limit, examples, properties of limit, and also some special
functions of limits

**Definition of limit:**

Let a function f (x) can be defined in an open
interval near the number “a “, (need not be at a).

If, as x approaches a specific number “L” then l
is called the limit of f(x) as x approaches a.

Symbolically it is a written Lim_{x}_{→}_{a }f(x) = L read as “limit of f(x), as x approaches
a.

**Types of limits:**

Generally, there are three types of limits

• Right-hand limit

• Left-hand limit

• Two side limits

**Right-hand limit**

A function f is said to have a right-hand limit if the Lim_{x}_{→}_{c }f(x) = M is read as the limit f(x) is equal to M
as x approaches c from the right.

**Formula _{}**

**Left-hand limit**

A function f is said to have a left-hand limit

Is read as the limit f(x) is equal to M as x
approaches c from the left hand.

**Formula**

**Two side limits**

Two side limits appear when the left- and
right-hand limit is equal.

Formula

=

The limit of the function is existing when the
left- and the right-hand limit is existing otherwise not exist.

**Properties of limits**

First of all, suppose that two functions Lim_{x}_{→}_{a }f(x) = S and Lim_{x}_{→}_{a }g(x) = P exists and c is any constant. Below are
a few properties of limit calculus that are helpful for determining limits problems.

**Addition property **

According to this property, the notation applied to
each function separately. The equation for the additional property is:

Lim_{x}_{→}_{a }{f(x)+g(x)} = Lim_{x}_{→}_{a }f(x)+ Lim_{x}_{→}_{a }g(x) = S + P

**For example
**

Lim_{x}_{→}_{3 }(2x+3x) = Lim_{x}_{→}_{3 }(2x) + Lim_{x}_{→}_{3 }(3x) (by using addition property)

Same as for subtraction.

**Constant property **

According to this property, the constant is
written outside the limit because the limit is only applied to the variable.
The equation for the constant property is:

Lim_{x}_{→}_{a }k f(x) = k Lim_{x}_{→}_{a }f(x) =KS

**For example**

Lim_{x}_{→}_{3 }(2x) =2 Lim_{x}_{→}_{3}(x) (by using constant property)

**Product property **

According to this property, a limit is applied to
each function separately. The equation for the product property is:

Lim_{x}_{→}_{a} f(x). g(x) = {Limx_{→}_{a} f(x)}. {Lim_{x}_{→}_{a }g(x)} = S.P

**For example**

Lim_{x}_{→}_{3}(2x). (3x) = {Lim_{x}_{→}_{3 }(2x)}. {Lim_{x}_{→}_{3 }(3x)} (by using product property)

**Quotient property **

According to this property, a limit is applied to
each function separately. The equation for the quotient rule i

Lim_{x}_{→}_{a }f(x)/g(x) = Lim_{x}_{→}_{a }f(x)/ Lim_{x}_{→}_{a} g(x) = S/P if P ≠ 0

**For example
**

Lim_{x}_{→}_{3} (2x)/(3x) = Lim_{x}_{→}_{3 }(2x)/ Lim_{x}_{→}_{3}(3x) (by using quotient property)

**Power property **

Lim_{x}_{→}_{a }{f(x)} ^{n} = {Limx_{→}_{a }f(x)}^{ n }= S^{n}

• Where **n**
is any integer

**The particular function of limit**

If, by substituting the number that x approaches
into the function, we get (0/0), then we evaluate the limit as follows:

We simplify the given function by using the
algebraic technique of making factors if possible and canceling the common
elements. The method is explained in the following limits.

• Lim_{x}_{→}_{a} (x^{n} – a^{n}) / (x – a) = na^{n-1}

_{•
}Lim_{x}_{→}_{a }sinx/x = 1_{}

• Lim_{x}_{→}_{0 }e^{x }= 1

_{•
}Lim_{x}_{→}_{+}_{∞}_{ }(1 + (1/n)) ^{n}=e (when n tends to
infinity through positive integral values only)_{}

• Lim_{ x}_{→}_{ }_{∞} (e^{x}) =_{ }∞

_{•
}Lim_{x}_{→}_{-}_{∞}_{ }(e^{x})=0_{}

_{•
}Lim_{x}_{→∞}_{ }(1 + (1/x))^{ x }=e (as x approaches
infinity while taking on or negative real values.)_{}

_{•
}Lim_{x}_{→∞}_{ }(a/x) =0_{}

Now we try to clear our topic by using some
simple and easy examples of limits.

**Solved Examples **

**Example: 1 **

Evaluate Lim_{x}_{→}_{3 }(9x^{2 }+ 2x + 3) if it exists.

**Solution:**

**Step
1****: ** first we apply the addition property on this
function so,

Lim_{x}_{→}_{3 }(9x^{2}) + Lim_{x}_{→}_{3}(2x) + Lim_{x}_{→}_{3} (3)

**Step
2:**

Now use the constant property of the limit and
write the constant outside of the limit.

9 Lim_{x}_{→}_{3}(x^{2}) + 2 Lim_{x}_{→}_{3 }(x) + Lim_{x}_{→}_{3}(3)

**Step
3:**

Now apply the limit on the function

= 9 (3^{2}) + 2(3) + 3

**Step
4: **by simplifying

81 + 6 + 3 = 90

**Example: 2**

Evaluate Lim_{x}_{→}_{3} (x-3/√x -√3) if it exists.

**Solution:**

If we apply the direct limit on the function, the
numerator, as well as the denominator, will become zero and we got the (0/0)
form so, first, we factorize the function and then apply the limit

**Step
1:**

By making the factors of (x-3)

Lim_{x}_{→}_{3} (√x -√3) (√x +√3) / Lim_{x}_{→}_{3} (√x -√3) (By using the quotient property)

**Step
2:**

Now the negative terms cancel each other and we
get only the positive term.

Lim_{x}_{→}_{3 }(√x +√3)

**Step
3:**

Now we apply additional property so,

Lim_{x}_{→}_{3} (√x) + Lim_{x}_{→}_{3}(√3)

**Step
4:**

Now apply the limit we get,

_{ }= (√3) + (√3)

= 2 √3

**Example 3:**

Evaluate Lim_{x}_{→∞}_{ }(1+1/9x)^{9x }if it exists.

**Solution:**

We know that Limx→∞ (1+1/x)^{x} = e by the particular
function of limit.

Lim_{x}_{→∞}_{ }(1+1/9x)^{9x} = e

**Conclusion:**

As we have
seen limit is a fundamental concept in mathematics and also for calculus. In
this article, we try to clear the definition, types, and some properties of
limit. And try to understand the topic through some examples. And also learn
how to solve the intermediate form of function.