Limit and Continuity, Objectives, Limit of Functions, Left and Right Hand Limits, a Limit of Function y = f (x) at x = a

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Objectives:

After studying this lesson, you will be able to:

  • Define limit of a function

  • Derive standard limits of a function

  • Evaluate limit using different methods and standard limits.

  • Define and interprete geometrically the continuity of a function at a point;

  • Define the continuity of a function in an interval;

  • Determine the continuity or otherwise of a function at a point; and

  • State and use the theorems on continuity of functions with the help of examples.

Limit of Functions:

In the introduction, we considered the function . We have seen that as approaches , approaches. In general, if a function approaches when approaches , we say that is the limiting value of

Symbolically it is written as

Example:

Find where

Solution:

We can solve it by the method of substitution. Steps of which are as follows:

We Can Solve It by the Method of Substitution
We can solve it by the method of substitution

Step 1:

We consider a value of close to a say

, where h is a very small positive number. Clearly, as

For we write , so

that as

Step 2:

Simplify

Now

Step 3:

Put and get the requried result

As

Thus,

by putting .

Left and Right Hand Limits:

  • You have already seen that means x takes values which are very close to , i.e. either the value is greater than or less than ’.

  • In case takes only those values which are less than and very close to ’ then we say is approaches from the left and we write it as .Similarly, if takes values which are greater than and very close to then we say is approaching from the right and we write it as .

  • Thus, if a function approaches a limit , as x approaches from left, we say that the left hand limit of as is .

We denote it by writing

or

Similarly, if approaches the limit , as approaches ’ from right we say, that the right hand limit of as is .

We denote it by writing

or

A Limit of Function Y=F(X) at X=A:

Consider an example:

Find , where

Solution:

Here

And

From (1) and (2),

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