examine the existence of limit?

Question

lim[x]
x....> root 2
i want to know how can the answer be 1

Answer 1

Your notation is a little unclear ... it looks like you are asking for which functions f(x) is it true that the limit as x approaches sqrt(2) of f(x) is 1. There will be many such f(x), the simplest perhaps is the constant function that maps every number to 1. Another easy choice is f(x) = x / sqrt(2) (or its reciprocal).

Dan

Answer 2

I assume by [x] you mean the greatest integer function.

Ok, pretend x gets closer and clsoer to sqrt(2), or, say, 1.414. Once you pass 1 on the number line (coming in from the left of 1.414), then [x] will always equal 1. So the limit from the left is 1.

Now, suppose you come in on the number line from the right of 1.414. Once you pass 2 (going to the left, towards 1.414), then [x] will also equal 1. So the limit from the right equals 1.

Since the left and right hand limits are qual, this limit equals 1.

Answer 3

If x is in the mod. there exists a +ive & a -ive value of x so, there will be left hand & right hand limit
= x =+1.414 & x= -1.414,
so the function will be continuous at 1 only .
Hence the answer is 1.


If this is correct or not reply me with a correct answer.
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