Calculus (LIMITS)

Question

Answer is 1. I also know the procedure. My question is why isn't this formula working?
Am I doing something wrong?

Comments

Check the formula..it is not right I think. The power will be g(x)*lnf(x)

https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/lHopital/power_limits.html

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@MiNiPanda  formula is correct in link

Answer 1

Tell me is this  1$^{\infty}$  form that you have applied this formula, the answer will be $1$ only, put $x=1/t$ then try to solve answer is $1$

see here
$\lim_{t \to 0}(\frac{1}{t})^{t} = \lim_{t \to 0}(\frac{1}{t^{t}}) = \frac{\lim_{t \to 0}(1)}{\lim_{t \to 0}(t)^{t}}  = 1$

Comments

My bad !! I spent so much figuring out the mistake :P
Thank you so much :)

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Or you can do it like this

$y = \lim_{x \to \infty} x^{1/x}$

Take 'log' on both sides. So this becomes

$log\ y = lim_{x \to \infty} \frac{log x}{x}$

Now using L Hospital's Rule on RHS

$log y = lim_{x \to \infty} \frac{1}{x}$

$log\ y = 0$

$y=e^0$

Therefore, $y = 1$