Limits at Infinity and Infinite Limits

Limits at Infinity and Infinite Limits

Limits at infinity

If x is a variable such that it can take any real value how much ever

If x is a variable such that it can take any real value how much ever

Example:

We write

Infinite limits

Let f(x) be a function of x, if the value of f(x) can be made greater than any pre-assigned number by taking x close to 'a', then we say

Similarly, if the value of f (x) can be made less than any pre-assigned number by taking x close to 'a', then we say f (x) tends to - as x approaches 'a'

Example:

In the first quadrant, as x approaches zero from the right of zero, the value of f(x) increases without bound. It becomes greater than any assigned positive real value.

In this case, we say

or

Similarly, in the third quadrant as x approaches zero from the left, the value of  decreases without bound. It can be smaller than any pre-assigned negative real number.

The following statement is useful to evaluate limits at infinity of rational functions.

Example:

Evaluate the limit:

Suggested answer:

One Sided Limit

We have discussed earlier about right hand limit and left hand limit. Both these limits are called one sided limits.

x approaches a from the right side and through values greater than a.

For a function f(x), we say

 as left hand Limit, as x approaches a from the left and through the values lesser than a.

The two important properties of these one-sided limits that

i) If the left hand limit and right hand limit of a function at a point exists, but are not equal, then we conclude that the limit at that point does not exist.

ii) If LHL and RHL of a function at a point (say a) exist and they are equal, we conclude that limit at that point exists and we write

We conclude our discussion on limits with one example on one sided limits.

Example:

For the function f(x) = x+ (x-[x])², find the RHL and LHL at x = 2. Check whether

Suggested answer:

RHL =

= 2

= 3