Limit Questions

[markraz]

Hi I have to solve the following limit:
\(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{x})^\frac{1}{x} \)

My question is, what is the order of operations here?
Should I plug \(\displaystyle \infty\) into the exponent first which gives me 0

then solve the base? or do I plug \(\displaystyle \infty\) into the base first?

1. \(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{x})^\frac{1}{\infty} \) = \(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{x})^0 \) = 1 (is this right?)

2. \(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{\infty})^0 \) = 1 (is this right? where I don't even need to solve the base)

or do I have to plug in \(\displaystyle \infty\) in the base? as follows??

if I solve the exp first I still get 1
3. \(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{\infty})^\frac{1}{\infty} \) = 1

or does #3 actually yield 0? as such
\(\displaystyle \Large\lim_{x \to \infty}(0)^0 \) = 0 (which is wrong I realize)

which of these 3 is the correct way to do it?

thanks

 

[Deleted member 4993]

Hi I have to solve the following limit:
\(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{x})^\frac{1}{x} \)

My question is, what is the order of operations here?
Should I plug \(\displaystyle \infty\) into the exponent first which gives me 0

then solve the base? or do I plug \(\displaystyle \infty\) into the base first?

\(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{x})^\frac{1}{\infty} \) = \(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{x})^0 \) = 1 (is this right?)

\(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{\infty})^0 \) = 1 (is this right?)

or do I have to plug in \(\displaystyle \infty\) in the base? as follows??

if I solve the exp first I still get 1
\(\displaystyle \Large\lim_{x \to \infty}( \frac{1}{\infty})^\frac{1}{\infty} \) = 1

however
\(\displaystyle \Large\lim_{x \to \infty}( 0)^0 \) = 0

y = \(\displaystyle \displaystyle{ln\left[\left(\dfrac{1}{x}\right)^{\frac{1}{x}}\right]}\)

= \(\displaystyle - \dfrac{ln(x)}{x}\) → 0 as x → \(\displaystyle \infty\)

\(\displaystyle \displaystyle{\lim_{x \to \infty} e^y = 1 }\)

 

[markraz]

thanks, so everything I had was wrong? or is your solution more prefered?