when both left and right hand limits tend to infinity can we say that the limit exists.Give reason(s).

We say that a function

f(x) "has a limit" L as x approaches , if for every sequence of values of x that approach as a limit -- whether from the left or from the right -- the corresponding values of f(x) approach L as a limit.

If that is the case, then we write:    "The limit of f(x) as x approaches l  is  L."

In other words, for the limit of f(x) to exist at a point x = l , the left-hand and right-hand limits must be equal.

Note:

iinfinite: If the absolute values of a variable (x or y) become and remain greater than any positive number we might name, however large, then we say that the variable "becomes infinite."

(1) As an example, here is the graph of the function :   y = 1/|x|

If x approaches 0 from the right , then the values of 1/|x| become large positive numbers( + infinity).

If x approaches 0 from the left, then the values of 1/|x| become large positive numbers( + infinity).

Now,when a limit is infinite, we say that the limit "exists."  which means that the definition of  "becomes infinite" has been satisfied.  Those left and right-hand limits therefore exist.
However, since they are same, the "limit as x approaches 0" does exist.

(2) As an example, here is the graph of the function :   y = 1/x

If x approaches 0 from the right , then the values of 1/x become large negative numbers( -infinity).

If x approaches 0 from the left, then the values of 1/x become large positive numbers( +infinity).

Here left and right-hand limits therefore exist but
However, since they are different, the "limit as x approaches 0" does not exist.

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Naga Ramesh
IIT Kgp - 2005 batch

YES we can say that the limit exists if both values come out to be infinity.Because limit of a function can attain a infinite value .This is so because in limits we say that the value is tending to infinty .However the value of a function can not attain infinity as because value of any function is always ABSOLUTE and thuscannot be infinity.

I hope its clear.