Determine the limit.
Lim (x-infinity) (1+sin(3/x))^(x)

Question

Determine the limit. 
Lim (x->infinity) (1+sin(3/x))^(x)

anonymous · Answer

$$\lim_{x \rightarrow \infty} (1+\sin \frac{ 3 }{ x })^{x}$$

hartnn · Answer

start by substituting 1/x = y
x =..?
when x tends to infinity, what value does y tend to ?

anonymous · Answer

I don't quite understand how you would substitute 1/x=y?
Did you mean 1/y?

hartnn · Answer

x = 1/y 
or 1/x = y
same thing

anonymous · Answer

So $$ \lim_{x \rightarrow \infty} (1+\sin(3y))^{\frac{ 1 }{ y }}$$ ?

hartnn · Answer

when x->infinity 
1/x ->0
y->0
got this ?

anonymous · Answer

Oohhh, okay. But the answer is $$e ^{3}$$ which I'm having trouble figuring out.

hartnn · Answer

ok, you know the formula 
lim x->0  (1+x)^(1/x) = e 
?

anonymous · Answer

Nope, how do I use that to solve this?

hartnn · Answer

ok, in that formula you see that its required that whatever is present with the "1+", should be present in the denominator of exponent. 

here we have sin 3x with "1+", so we need the exponent as 1/ sin 3x

hartnn · Answer

so, i'll adjust the exponent as 
1/y = 1/ sin 3y times sin 3y/ y
got this ?

anonymous · Answer

Yea!
thanks for the explanation

hartnn · Answer

need further explanation ?

anonymous · Answer

yes please, if that's okay

hartnn · Answer

sure,
now we will take the limit of exponent separately 
|dw:1383720849532:dw|got this ??