Verified answer

Answer:

1/4

Step-by-step explanation:

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I'm Naira.... you????

Answer: 1/4

Explanation:

You can solve this in 2 ways:

(i) We know that sin(2x) = 2(sinx)(cosx) ⇒ sinx = sin(2x)/(2cosx)

So, sin(x/4) = sin(x/2)/2cos(x/4) = sinx/4cos(x/2)cos(x/4)
⇒ sin(x/4)/x = sinx/4xcos(x/2)cos(x/4)

Here we can replace (sinx)/x as 1 since lim x→0

Now you can apply the limit to 1/4cos(x/2)cos(x/4) and you get
1/4cos(0)cos(0) = 1/4

(ii) Since this is in the form 0/0 you can apply L'Hospital rule here

d/dx{sin(x/4)} = cos(x/4)/4
d/dx(x) = 1

So the expression becomes,

lim x→0 cos(x/4)/4

Now you can just apply the limit and replace x with 0

cos(0/4)/4 = cos(0)/4 = 1/4

Note: As a general rule you can just remember: lim x→0 sin(x/n)/x = 1/n

Hope this helped