Question

Lim (x⇒0) (tg³2x/sin²3x)

Answer 1

умножим числитель и знаменатель дроби на выражение (2х)^3*(3x)^2

=lim(x→0)sin^3(2x)*(2x)^3*(3x)^2/(sin^2(3x)*cos^3(2x)*(2x)^3*(3x)^2)

воспользуемся первым замечательным пределом и учтем, что

lim(x→0)cos^3(x)=1; получим lim(x→0)(2x)^3/(3x)^2=lim(x→0)(8x/9)=0

Answer 2

To find the limit of the given expression as x approaches 0, we can use some trigonometric identities and algebraic manipulations.

Given expression: lim(x⇒0) (tan³(2x) / sin²(3x))

Step 1: Simplify the expression

We know that tan(x) = sin(x) / cos(x). Using this, we can rewrite the expression as follows:

lim(x⇒0) ((sin(2x) / cos(2x))³ / sin²(3x))

Step 2: Apply the identity sin(2x) = 2sin(x)cos(x)

lim(x⇒0) (((2sin(x)cos(x)) / cos(2x))³ / sin²(3x))

Step 3: Cancel out common terms

lim(x⇒0) ((2sin(x)cos(x))³ / cos³(2x) * sin²(3x))

Step 4: Apply the identity cos(2x) = 1 - 2sin²(x)

lim(x⇒0) ((2sin(x)cos(x))³ / (1 - 2sin²(x))³ * sin²(3x))

Step 5: Use the trigonometric identity sin(3x) = 3sin(x) - 4sin³(x)

lim(x⇒0) ((2sin(x)cos(x))³ / (1 - 2sin²(x))³ * (3sin(x) - 4sin³(x))²)

Step 6: Simplify further

Now, let's expand the expression and simplify by canceling out common factors:

lim(x⇒0) ((2sin(x)cos(x))³ / (1 - 2sin²(x))³ * (9sin²(x) - 24sin⁴(x)))

Step 7: Substitute x = 0 into the expression

Now, we can directly substitute x = 0 into the expression:

lim(x⇒0) ((2sin(0)cos(0))³ / (1 - 2sin²(0))³ * (9sin²(0) - 24sin⁴(0)))

Since sin(0) = 0 and cos(0) = 1, the expression becomes:

lim(x⇒0) (0³ / (1 - 2 * 0²)³ * (9 * 0² - 24 * 0⁴))

lim(x⇒0) (0 / (1 - 0)³ * 0)

lim(x⇒0) (0)

Step 8: Final result

The limit of the given expression as x approaches 0 is 0:

lim(x⇒0) (tan³(2x) / sin²(3x)) = 0