How to solve this limit #\lim_{x\to pi/4} (sin2x)^(tan^2(2x)) # ?
2 Answers
Explanation:
you can Look at this link for Reference https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/lHopital/power_limits.html
The trick here is to use L'Hopital's Rule. although you might say there are no fractions but we can convert them into it. here is the start of my solution
And I want to clarify, I will be using the variable which you used as y and not x and use x as equal to 2y
therefore your question is
to simplify,
let
Realize that the equation is equal to
as When
Now, Break this up into an exponential with the base e
therefore,
using exponential rules
now, Bring the limit inside the equation using the Composition Law
https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/lHopital/limit_laws.html#Composition_Law
therefore the above equation is equal to
Now Lets just focus on the Limit and try to use L'hôpital's Rule on it, but first, we need to convert it into a division of two functions
therefore,
I assume you already know how to take the derivative of the two functions
after cancelling cot (x), we Get
therefore the limit is
which is continuous and defined at
and don't forget, this is the power
Explanation:
Making
now
but
but
then
because
Now expanding