# Some fundamental of limits

I am studying the limit chapter of calculus...In some earlier examples I found that if a function with certain limit results into more than two values then limit DNE.

I am wondering if we evaluate RHL and LHL. They both are finite but do not equal. Then does the limit still exist?? I mean , here too, we get two different numbers.

And one more thing I am doubtful about is when should we exactly use the LHL and RHL approach.... Because there are still more convenient ways to find limits.

When RHL and LHL exist but are not equal, you can say that each of the two limits exist in that point.

EG

$\frac{1}{x}\to+\infty$ as $x\to 0^+$

$\frac{1}{x}\to-\infty$ as $x\to 0^-$

As a simple example of limit which not exists at all consider:

$\sin x$ as $x\to +\infty$

To deal with this kind of limit we can define the concept of supremum limit and infimum limit (which are respectively +1 and -1 for $\sin x$ as $x\to +\infty$).

• When should I use the LHL and RHL approach?? I reckon one is when we r dealing with piece wise function and?? – demon Dec 16 '17 at 10:28
• In many cases you may be interested to study the behavior of the function only at one side, as an example to study the graph of a function or if yu are dealing with function for which the limit is defined only at one side (EG $\log x$ as $x\to 0^+$). – user Dec 16 '17 at 10:44
• understood. Thanks. – demon Dec 16 '17 at 16:57