This is the first thing that occurs to me so a better answer may come along but I would operate on the basis that
from the formal definition
The red and blue terms are well known limits provable using squeeze theorem.
Now to prove that
if we start with:
But by the chain rule applied to
That completes this approach, I think.
Refer to the Explanation Section below.
To evaluate this Limit, let,
Hence, in this new scenario,
Here, we see that, the