how do u prove the uniqueness of a limit?

Answer 1

You'll need the formal definition of a limit.
I won't go through all the gritty details, but the process will be something like this:
Assume there are two limits. Then there must be some gap between those limits - so lets take a small number ε which is less than half the gap between those limits.

Now, by definition of a limit, as we get closer and closer to the x value, the function should get closer and closer to the limit. Eventually we will get so close that we need to be no more than ε away from each limit. However, thats impossible - since the distance between limits is more than 2ε, theres no way to be within ε of both.

Now, its up to you to turn that into a formal proof :) Its not that hard. Let me know if you get stuck doing it.

Answer 2

For limits of sequences:

Suppose the sequence a_n has two different limits A and B.

Then any environment of A contains all terms of a_n after an index N_a, depending on the radius of the environment, and any environment of B contains all terms of b_n after an index N_b, which depends on the radius of the environment. Then all terms after the index max(N_a, N_b) will be in both the selected environment around A and the environment around B, so they will be in the intersection of the two environments. However, because A and B are different, you can take environments around them which are disjoint, and the intersection of the environments is supposed to contain all terms of a_n after the index max(N_a,N_b), however, it contains no terms, because it is the intersection of two disjoint sets, which is empty.

This is a contradiction which resulted from the assumption that a_n had two different limits.

(By an environment of A, I mean an interval of the form (A-epsilon, A+epsilon), where epsilon>0 is the radius of the environment.)

For limits of functions:

A function f(x) has a limit equal to A at x=a if and only if for any sequence a_n converging to a, the sequence f(a_n) converges to A (it is a theorem). Thus, limits of functions can be defined in terms of limits of sequences, which are unique by the above reasoning, so limits of functions are also unique.