Question

What is the value of x→0 {Sin(3x)/SIn(4x) without using L'Hopital's rule?

a) 0
b) 1/8
c) 1/4
d) 1/2
e) None of the above

Answer 1

The value of x→0 {Sin(3x)/Sin(4x)} without using L'Hopital's rule is 1/4.

Step-by-step explanation:

To find the value of x→0 {Sin(3x)/Sin(4x)} without using L'Hopital's rule, we can use the basic properties of sine. First, we can rewrite the expression as (Sin(3x))/(Sin(4x)). Then, using the identity Sin(3x) = 3Sin(x) - 4Sin³(x) and Sin(4x) = 4Sin(x) - 8Sin^3(x), we can simplify the expression to (3Sin(x) - 4Sin³(x))/(4Sin(x) - 8Sin³(x)).

Next, we can divide every term in the numerator and denominator by Sin(x) to get (3 - 4Sin²(x))/(4 - 8Sin²(x)).

Finally, as x→0, Sin(x)→0, so the value of the expression becomes (3 - 4(0))/(4 - 8(0)) = 3/4. Therefore, the correct answer is c) 1/4.