This morning a colleague asked me how I would solve quickly this limit, lim x...
This morning a colleague asked me how I would solve quickly this limit,
Answer & Explanation
Consider rewriting the numerator as Then we can use the same manipulation to get
Substituting this in gives us
supposing for the moment that those two limits exist. (also note the additional factor of 2 in the denominator which seems to have been dropped in your manipulation)
For the first limit we can simply evaluate at to get
For the second limit, consider rewriting in the numerator as This gives us that
So our total limit is which agrees with your initial solution.
If you set with then you get
The fact that can be inferred from: