l’Hopital’s rule takes care of both of those in one step.
Prove that when c ∈ N:
a)
lim [(1+x)^(1/c) - 1]/x = 1/c
x->0
b)
lim [(1+x)^r - 1]/x = r
x->0
,where r = c/n
My approach for a and b are pretty similar, i get stuck at the same point.
For a, i multiplied the numerator and the denominator by [(1+x)^(1/c) + 1] to get rid of the radical on top. This left me with:
Lim 1/[(1+x)^(1/c)+1]
x->0
This is where im stuck, if i find the limit of the numerator and divide it by the limit of the denominator, i dont get 1/c. So im clearly doing something wrong.
As for part c, i pretty much use the same approach and get:
Lim 1/[(1+x)^(r)+1]
x->0
which also doesnt lead to the answer.
Please help.