Left-Hand Limits and Right-Hand Limits - Practice Questions & MCQ

Left-Hand Limits and Right-Hand Limits

With continuing from previous concept, we can approach the input of a function from either side of a value—from the left or the right.  

we had our function as

  which becomes equivalent to the function

now, let us observe the values of f(x) nearby x = 7.

Approaching 7 “from the left” means that the values of input are just less than 7. And if for such values of x, the values of f(x) are close to L, then L is called the left-hand limit of function at x = 7. For this function, 8 is the left-hand limit of the function f(x) at x = 7.

Approaching 7 “from the right” means that the values of input are just larger than 7. And if for such values of x, the values of f(x) are close to R, then R is called the right-hand limit of function at x = 7. For this function, 8 is the right-hand limit of the function f(x) at x = 7.

To indicate the left-hand limit, we write . 7^- indicates the values that less than 7 and are infinitesimally close to 7

To indicate the right-hand limit, we write. 7⁺ indicates the values that greater than 7 and are infinitesimally close to 7

The left-hand and right-hand limits are the same for this function.

( It denotes that if we can redefine the function, it is possible to define value of function at x = 7 as f(7) = 8... this is the scenario we wiil be observing the the chapter again and again where, we define it that if LHL = RHL at a point x=a, then Limit of function at the point can be defined)

The left-hand limit of a function f(x) as x approaches a from the left is denoted by 

The right-hand limit of a function f(x) as x approaches a from the right is denoted by 

Now consider a function,  

Let’s check the behavior of f(x) in the neighborhood of x = 0

Existence of a limit of a function

From the above example, we can define the existence of limit

i.e., LHL = RHL at x = a 

Also notice that the limit of a function can exist even when f(x) is not defined at x = a.