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Find lim┬(x→0+)⁡〖x^x 〗. (personal note: I think the approach is to use l'Hopital's rule)
Prove that e is irrational. (hint: Suppose false, so that e = p/q where p,q ∈ N. Write e = e^1= P_n (1)+ R_n (1), multiple both sides by n! and deduce a contradiction when n ∈ N is sufficiently large).
Expand the polynomial p(x)= 〖3x〗^3+〖2x〗^2- x+1 as a polynomial in powers of (x-1): That is, show that p(x)= ∑_(k=0)^3▒c_k 〖(x-1)〗^k and find the values of the constants c_(0 ),...,c_3

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1. Find . (personal note: I think the approach is to use l'Hopital's rule)

Solution:

Let y =

Take log on each side

lny = ln( )

lny =

We can write it as

lny =

Now apply L'Hospital's rule

lny =

lny tends to 0.

Therefore y will tend to e^0 = 1.

2. Prove that e is irrational. (hint: Suppose ...

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