Limits In Calculus

Before talking about limits in Calculus, one must be familiar with few basic topics of calculus like Functions, range and Domain. These are very important to understand the concept of Math because these are the basics requirement for studying calculus limits. So in calculus you can say that a function’s behavior is called limit of that function.

Now, let’s see what Mathematical Expression of limit is. General notation of Limits in Calculus is given as:

lim _x→c f(x)  =L,

Where, L the limit of f(x) as x approaches c if f(x) becomes close to L when x is closed to c. If there is no other value of L in the same condition

lim _x→c f(x)  =L,

Or,

f(x) → L as x → c,

The definition of a limit is not concerned with value of f(x) when, x=c. So, we care about the values of f(x) when x is close to c, on either the left side or right side.

Now, we move on to rules regarding limits. Limits have some rules which are useful when we solve different limit problems.

First Rule: First rule is called as the constant rule. In this rule we state- if we have f(x) =b (where f is constant for all x) then, the limits as x approaches c must be equal to b.

It means limit of function appears as shown below:

If b and c are constant then, our limit is lim _x→c  b  =b.

Second Rule: Second rule is called the identity rule. 

 If f(x) =x,

Then, the limit of f as x approaches c is equal to c.

According to this, the function looks like:

If c is a constant then lim _x→c x =c.

Here are some Operational Identities for limits, which are given below:

 Suppose we have two limit Functions which are

lim _x→c f(x) =L and lim _x→c g(x) =M  and k is constant value of function

Then the limit is:

lim _x→c kf(x)  = k. lim _x→c f(x)  =kL

lim _x→c [f(x) +g(x)] = lim _x→c f(x)  + lim _x→c g(x)  =L+M

lim _x→c [f(x) -g(x)] = lim _x→c f(x)  - lim _x→c g(x)  =L-M

lim _x→c [f(x)g(x)] = lim _x→c f(x)  lim _x→c g(x)  =LM

lim _x→c [f(x) /g(x)] = lim _x→c f(x)  / lim _x→c g(x)  =L/M (where M is not equal to zero .)

Let’s see the examples based on formulas.

Example: Find the limit of this given function

lim _x→2 3x⁴?

Solution: First, we have to specify the problem as we have no rule of the expression by this function. We know the identity rule from above that lim _x→2 x =2

By this rule,

lim _x→2 x⁴ =( lim _x→2x )⁴ = 2⁴ =16,

By Scalar multiplication rule we get the function as:

lim _x→2 3x⁴  =3 lim _x→2 x⁴  = 3 x 16 =48.

Example: Solve the limit function where,

lim _x→2 x³ + x² +1?

Solution: For solving this limit function we have to follow the expression given below.

Step 1: In first step we write the given limit function,

lim _x→2 x³ + x² +1,

Step 2 : In second step we have solved particular values,

lim _x→2 x³ + lim _x→2 x² +lim _x→2 1,

lim _x→2(2)³ + lim _x→2 (2)² +1,

8 + 4 +1 = 13,

Now we get the value of limit of function is 13.

Let’s see example of limit in simple approach to infinity,

Y=3x,

As,  x=1,  y=3,

X=2 y=6,

X=3 y=9,

X=4 y=12,

And so on,

X=100  y=3000,

Now, x and y both approaches to infinity.

Let's have one more example in order to understand this, 

2x² -5x,

2x² will always tend towards infinity and -5x always tends towards minus infinity if, 'x' will increase where will the function tends?

It will always depend on the value of if x² will grow more rapidly with respect to x as x increases then the function will surely tend towards the positive infinity

Now, let’s talk about the degree of the function, it can be defined as the highest power of variable for example:

5x² +6x+7,

In this x² has highest power as two so degree of the function will be 2. The degree of the function can be negative or positive. If degree of the function, is greater than 0 then, limit will always be positive. If degree of the function is less than 0 then the limit will be 0.