As a curiosity, I’ll calculate the limit of the function:
More interesting than the calculations themselves are the two morals that can be drawn from them. But the morals are at the end (has anyone ever seen a story with a moral at the beginning)? Now, I’m calculating:
At the beginning, I look at what the expression approaches, and I get:
So, I have no idea what it approaches because I don’t even know what 
approaches (this is an indeterminate form), let alone something weird like: 
.
However, it looks like there are no problems with the cosine, so I calculate on the side what this approaches:
I will use the approach of L’Hôpital (I showed exactly what this is about in my Course), which means I’ll use the formula (
):
But it’s still not great, because I still have an indeterminate form in the exponent 
, so I transform it a second time:
Now I can calmly get to work. I calculate on the side:
, so I have an indeterminate form 
so I use L’Hôpital’s rule:
So, I’ve calculated on the side that: 
.
Returning to the limit:
I now know that I’m in the situation:
So it turns out that:
So, my whole limit is:
And the cosine was just for show.
Morals of this story
There are two.
- In harder limits, you can always and sometimes need to take a part of it and calculate what it approaches “on the side”.
- Sometimes, some parts of the limit formula can be added in like this cosine. It turned out that I didn’t have to do anything with it, just substitute zero at the end.
