# 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