Limits And Continuity

Have you ever driven a motorbike or a car?

What do we observe when we press the accelerator for any one of them?

The motorbike/car attains its maximum speed after some time int`erval say 10-15 seconds.

Now, the speed might increase from 0 km/hr initially i.e. at time 0 seconds to 100/150 km/hr after time 10-15 seconds. Now, if we are interested in finding the approximate speed of the motorbike/car at the time say 8 seconds, then we want an expected or estimated value of the speed at a particular time instant.

Here, comes the role of limits.

Similarly, if we are not able to find or determine the actual value of a function say f(x) at any given point say x = a, we try to estimate i.e. find its expected value at x = a. This is where the concept of limit comes in picture.

Limit of a Function

Let f(x) be function and x = a be any point in its domain then the limit of f(x) at x = a is denoted by f(x) . It is the expected or estimated value of f(x) as x approaches to a.

• Left Hand limit: The left hand limit of f(x) at x = a is denoted by  f(x). It is the expected or estimated value of f(x) at x = a when the values of f(x) near to and to the left of a are given.
• Right Hand limit: The right hand limit of f(x) at x = a is denoted by f(x). It is the expected or estimated value of f(x) at x= a when the values of f(x) near to and to the right of a are given
• Existence of limit: If the left hand limit & right hand limit both coincide i.e. are equal then the limit exists & the common value is called the limit of the function.

If   f(x) =  f(x) = l.(say) 

Then,  f(x) exists &  f(x) = l.

We read it as the limit of f(x) as x approaches to a is equal to l.

 Standard Limits

1.  Limit of polynomial function, f(x) = a₀ + a₁x + a₂x² + a₃x³ + ….+ a_nxⁿ

f(x)= f(a)

= (a₀ + a₁x + a₂x² + a₃x³ + …. + a_nxⁿ)

= (a₀ + a₁a + a₂a² + a₃a³ + …. + a_naⁿ)

2. Limit of Rational function  =  =  provided g(x)0.
   
3.  =n

Algebra of limits of functions

If f(x) and g(x) be any two functions such that  f(x) & g(x) both exist.

1. [f(x)+g(x)] =  f(x) +  g(x)
2. [f(x) – g(x)] =  f(x) – g(x)
3. [f(x) g(x)] =  f(x)g(x)
4.  =   where g(x)  0 .

 Examples

 Now let’s consider some examples on limits and continuity.

Example 1: Findf(x) where f(x) = x¹¹ + 3x.

f(x) = (x¹¹ + 3x) = (1)¹¹ + 3(1) = 4

 Example 2: Find f(x) where f(x) =  .

f(x) =  = 

=  = 

Example 3: Find  f(x) where f(x) =  .

 f(x) =  = 10 = 10 using  = n

Continuity

A function f(x) is called continuous at a point x = a in its domain if f(x)= f(a), which can also be stated as

f(x) =f(x) = f(a).

Hence, if the left hand limit & right hand limit both exist, and are both equal to the value of f(x) at x = a, then the function f(x) is called continuous at a point x = a.

Example 4: Check the continuity of the function f(x) = x¹¹ + 3x at x = 1.

 f(x) =  = (1)¹¹ + 3(1) = 4

f(1) = (1)¹¹ + 3(1) = 4

Therefore,  f(x) = f(1). 

Hence, the function f(x) = x¹¹ + 3x is continuous at x = 1.

 Example 5: Find the value of a if f(x) = 2x + a is continuous at x = 1 and f(1) = 5.

Since f(x) = 2x + a is continuous at x = 1,

∴ f(x)=f(1)

(2x+a)=5 

2(1)+a=5

a=5-2=3

Check Point

1. Find f(x) when f(x)  = x² + 2x + 3.
2. Find  f(x) when f(x) =  . 
3. Find  f(x) when f(x) = 
4. Check the continuity of the function f(x) = 3x² + 5x at x = 2.
5. Find the value of a if f(x) = 3x² + a is continuous at x = 2 and f(2) = 17.

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