Here how about this,

Let and be two functions such that

Or in other words

Then you can replace with in any non-related limit that goes to c.

Here I will show why

Say we are still talking about

But we need to compute the limit

And if we could replace with this would be much simpler.

So here is how we can see we can

Since

From our laws of limits we see that

Assuming both limits exist, we then solve for the limit in question

.

Now you may be wondering how this helps, but once again consider that our limit we wish to compute may be rewritten as

Now we see by the equation above that we may rewrite this as

Which is what we desired.

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So now for the first example what you may do is this

Since

By the substitution

and similarly for

We may say that

and

So we may rewrite our limit as follows

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Similarly by a substitution of you can show that

Or just consider that

and as

So we can see that

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The last limit does not exist, to make it a little more clear we can see that

since

So

So we may rewrite this as

In which case it should be apparent the limit does not exist since

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Note the first two limits could have also been done as follows

Now rewriting this limit as the products of its individual components we see that we have

The last two limits can be found by making the sbustitutions and respectively.

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and also we could rewrite

Now rewriting it as

where the aforementioned substitution should be made.