How do you solve this limit of a function math problem?

If you know that

then it's possible to rewrite the given limit so that it resembles the one above. Then the limit itself would be some expression involving .

For starters, we have

Let . Then as , we also have , and

So in terms of , the limit is equivalent to

Now use some of the properties of limits: the above is the same as

The first limit is trivial; , so its value is 1. The second limit comes out to

To see why this is the case, replace , so that as , and

Then the limit we're talking about has a value of

# # #

Another way to do this without knowing the definition of as given above is to take apply exponentials and logarithms, but you need to know about L'Hopital's rule. In particular, write

(where the notation means , just to get everything on one line).

Recall that

if is continuous at . is continuous everywhere, so we have

For the remaining limit, write

Now as , both the numerator and denominator approach 0, so we can try L'Hopital's rule. If the limit exists, it's equal to

and our original limit comes out to the same value as before, .

think about it deeply and passionately about this property

i am sorry