In general how would i simplify these expressions showing enough proof? and is there a website where i see all the rules for limits?
1. Limit Sin ax/Sin bx
x -->0
2. Limit Tan ax/bx
x -->0
3. Limit Cos x/x
x -->0
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.