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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.

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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 $x\to +\infty$).

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  • $\begingroup$ When should I use the LHL and RHL approach?? I reckon one is when we r dealing with piece wise function and?? $\endgroup$ – demon Dec 16 '17 at 10:28
  • $\begingroup$ 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 $x\to 0^+$). $\endgroup$ – user Dec 16 '17 at 10:44
  • $\begingroup$ understood. Thanks. $\endgroup$ – demon Dec 16 '17 at 16:57

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