Find-the-exact-value-of-0-0-

Question Number 48291 by Rio Michael last updated on 21/Nov/18

F i n d t h e e x a c t v a l u e o f 0^{0}

Commented by maxmathsup by imad last updated on 21/Nov/18

0^{0} = {l i m}_{x \to 0^{+}} x^{x} = {l i m}_{x \to 0^{+}} e^{x l n (x)} = e^{0} = 1 b e c a u s e {l i m}_{x \to 0^{+}} x l n (x) = 0 f o r

t h a t w e t a k e 0^{0} = 1 .

Answered by MJS last updated on 21/Nov/18

0^{0} is not defined

1^{st} step

a \div b = \frac{a}{b} = c \Rightarrow a = c \times b

\Rightarrow b \neq 0 because if b = 0 we have

a \div 0 = \frac{a}{0} = c \Rightarrow a = c \times 0 = 0

example : 5 \div 0 = \frac{5}{0} = c \Rightarrow 5 = c \times 0 = 0 which is wrong

conclusion : a \div 0 = \frac{a}{0} is not defined \Rightarrow

\Rightarrow a \div b = \frac{a}{b} = c for b \neq 0

2^{nd} step

a^{n} = a \times a \times a \times \dots (n times)

0^{n} = 0 \times 0 \times 0 \times \dots = 0

\frac{a^{m}}{a^{n}} = a^{m - n} \Rightarrow \frac{a^{n}}{a^{n}} = a^{0} = 1 but because of our

above conclusion a \neq 0 \Rightarrow 0^{0} is not defined

but sometimes it makes sense to define it

for some reasons . in these cases we are free

to set the value . f (x) = \frac{x}{x} = 1 \forall x \neq 0 \Rightarrow we

can define f (0) = 1 to keep the function

continous . or f (x) = x^{x} . usually we want to

give it the value of the limit .

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