Evaluate the limit
lim sin4x/tan9x
x-> 0
To evaluate the given limit, we can simplify the expression by using trigonometric identities.
The trigonometric identity that relates sine and tangent is:
tan(x) = sin(x) / cos(x)
Using this identity, we can rewrite the expression as:
lim (sin(4x) / (tan(9x))
x->0
= lim (sin(4x) / (sin(9x) / cos(9x)))
x->0
= lim (sin(4x) * cos(9x) / sin(9x))
x->0
Now we can simplify further by using the trigonometric identities:
sin(2a) = 2sin(a)cos(a)
Applying this identity to the numerator, we get:
sin(4x) = 2sin(2x)cos(2x)
The expression now becomes:
lim ((2sin(2x)cos(2x)) * cos(9x) / sin(9x))
x->0
The sin(9x) term cannot be simplified further, so we will focus on the other terms.
lim ((2sin(2x)cos(2x)) * cos(9x) / sin(9x))
x->0
= 2 * lim (sin(2x)cos(2x) * cos(9x) / sin(9x))
x->0
The sin(2x) term can be rewritten as:
sin(2x) = 2sin(x)cos(x)
The expression now becomes:
2 * lim ((2sin(x)cos(x))cos(2x) * cos(9x) / sin(9x))
x->0
= 4 * lim (sin(x)cos(x)cos(2x) * cos(9x) / sin(9x))
x->0
Finally, we can apply the trigonometric identity:
sin(2a) = 2sin(a)cos(a)
Using this identity, we rewrite the expression as:
= 4 * lim (sin(x) * (2cos(x)cos(2x)) * cos(9x) / sin(9x))
x->0
= 4 * lim (sin(x) * (cos(3x) + cos(x)) * cos(9x) / sin(9x))
x->0
Now, we can see that as x approaches 0, all terms in the numerator and denominator will approach 0 except for the cos(9x) term. Therefore, we can simplify the expression further:
= 4 * lim (cos(9x))
x->0
Now, we can evaluate the limit:
lim (cos(9x))
x->0
= cos(9*0)
= cos(0)
= 1
Therefore, the limit of sin(4x) / tan(9x) as x approaches 0 is equal to 1.