QuestionFind the limit: $\lim _{x \rightarrow 0}(\cot (6 x) \cdot \sin (2 x))$ .

Studdy Solution

STEP 1

Assumptions1. We are given the limit problem $\lim{x \rightarrow0}(\cot (6 x) \cdot \sin ( x))$ . . We need to find the limit as $x$ approaches $0$ .
3. We will use the limit properties and trigonometric identities to solve the problem.

STEP 2

First, we need to rewrite the cotangent function as the reciprocal of the tangent function.
$\lim{x \rightarrow0}(\frac{1}{\tan (6 x)} \cdot \sin (2 x))$

STEP 3

Next, we rewrite the sine function as $2 \sin x \cos x$ using the double angle formula.
$\lim{x \rightarrow0}(\frac{1}{\tan (6 x)} \cdot2 \sin x \cos x)$

STEP 4

We can now rewrite the limit problem as a product of two limits.
$\lim{x \rightarrow0}(\frac{2 \sin x \cos x}{\tan (6 x)})$

STEP 5

We can now rewrite the limit problem as a quotient of two limits.
$\lim{x \rightarrow0}(\frac{2 \sin x \cos x}{\sin ( x)/\cos ( x)})$

STEP 6

We can now simplify the limit problem.
$\lim{x \rightarrow0}(2 \sin x \cos x \cos (6 x)/\sin (6 x))$

STEP 7

We can now apply the limit properties to the limit problem.
$2 \lim{x \rightarrow0}(\sin x \cos x \cos (6 x)/\sin (6 x))$

STEP 8

We can now apply the limit property $\lim{x \rightarrow a}(f(x)/g(x)) = \lim{x \rightarrow a}f(x) / \lim{x \rightarrow a}g(x)$ if $\lim{x \rightarrow a}g(x) \neq0$ .
$2 \lim{x \rightarrow0}(\sin x \cos x \cos (6 x)) / \lim{x \rightarrow0}(\sin (6 x))$

STEP 9

We can now apply the limit properties to the limit problem.
$2 \lim{x \rightarrow}(\sin x) \lim{x \rightarrow}(\cos x) \lim{x \rightarrow}(\cos (6 x)) / \lim{x \rightarrow}(\sin (6 x))$

STEP 10

We can now evaluate the limits.
$2 \cdot0 \cdot \cdot /0$

STEP 11

The limit is undefined because we have a form of $0/0$ which is an indeterminate form.So, we need to apply L'Hopital's rule which states that if the limit of a function leads to an indeterminate form ( $0/0$ or $\infty/\infty$ ), then the limit of that function is equal to the limit of the derivative of the numerator divided by the derivative of the denominator.

STEP 12

Differentiate the numerator and the denominator.
$\lim{x \rightarrow0}(\frac{2 \cos x -12 \sin x \cos x}{6 \cos (6 x)})$

STEP 13

We can now evaluate the limit.
$\lim{x \rightarrow0}(\frac{2 \cos x -12 \sin x \cos x}{6 \cos (6 x)}) = \frac{2 \cdot -12 \cdot0 \cdot}{6 \cdot} = \frac{2}{6} = \frac{}{3}$ So, the solution to the limit problem $\lim{x \rightarrow0}(\cot (6 x) \cdot \sin (2 x))$ is $\frac{}{3}$ .

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