Sum of 0^0

Calculus Level 3

let a, b and c be real numbers such that $a=\lim_{x->0}x^{0}, b=\lim_{x->0^{+}}0^{x},$ $c=\lim_{x->0}x^{x}$ find $a+b+c$

1 solution

Nelson Mandela
Jul 4, 2015

a = lim(x tends to 0)x^0 that implies x is very close to 0 like 0.001 and any number to the power 0(except 0) is 1 so a=1.

b = lim(x tends to 0)0^x. that implies x is very close to 0 like 0.001 and 0 to the power anything (except 0) is 0 so b=0.

now c is confusing for me because as per me it is 0.001^0.001 which is close to zero and the c should have been 0 but,

x^x = e^xln(x). now if we consider the lim(x tends to 0)xln(x),

it can be written as ln(x)/(1/x).

using l'hopital rule,

lim(x tends to 0)f(x)/g(x) = lim(x tends to 0)f'(x)/g'(x) if lim(x tends to 0)f(x)/g(x) is 0/0 or -infinite/infinte.

here it is -infinite/infinite because e^-infinite = 0(x). and 1/0(x) = infinite.

so, c is e^lim(x tends to 0) (1/x)/(-1/x^2) = e^lim(x tends to 0) -x = e^0 = 1.

so, c = 1 and a+b+c=1+0+1=2.