# Limit and Continuity, Basic Theorem on Limits, Finding Limits of Some of the Important Functions

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## Basic Theorem on Limits

being a constant.

, provided

We have seen above that there are many ways that two given functions may be combined to form a new function. The limit of the combined function as can be calculated from the limits of the given functions. To sum up, we state below some basic results on limits, which can be used to find the limit of the functions combined with basic operations.

If and , then

(i) where is a constant.

(ii)

(iii)

(iv) , provided

The above results can be easily extended in case of more than two functions.

Example:

Evaluate:

Solution:

Example:

Find , if it exists.

Solution:

We choose values of that approach from both the sides and tabulate the corresponding values of .

We see that as , the corresponding values of are not getting close to any number.

Hence, does not exist. This is illustrated by the graph in Fig.

## Finding Limits of Some of the Important Functions:

**(1) Prove that** **where n is a positive integer.**

Proof:

**(2) Prove that (a)** **and (b)**

Proof:

Consider a unit circle with centre , in which is a right angle and radians.

Now C and

As x decreases, A goes on coming nearer and nearer to .

i.e., when

Or when

And ,i.e.,

When and

Thus we have

and