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

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches0 for the function $\frac{\sin (6 x)}{\sin ( x)}$ . . We will use the limit properties and trigonometric identities to solve this problem.

STEP 2

We notice that the limit is of the form $\frac{0}{0}$ as x approaches0. This is an indeterminate form, so we can apply L'Hopital's rule, which states that the limit of a ratio of two functions as x approaches a certain value is equal to the limit of the ratios of their derivatives.
$\lim{x \rightarrow0} \frac{\sin (6 x)}{\sin (2 x)} = \lim{x \rightarrow0} \frac{(6 \cos (6 x))'}{(2 \cos (2 x))'}$

STEP 3

Now, we calculate the derivatives of $\sin (6 x)$ and $\sin (2 x)$ using the chain rule. The derivative of $\sin(u)$ is $\cos(u) \cdot u'$ , where $u'$ is the derivative of $u$ .
For $\sin (6 x)$ , $u =6x$ and $u' =6$ . So, the derivative is $6 \cos (6 x)$ .
For $\sin (2 x)$ , $u =2x$ and $u' =2$ . So, the derivative is $2 \cos (2 x)$ .

STEP 4

Substitute these derivatives back into the limit.
$\lim{x \rightarrow0} \frac{6 \cos (6 x)}{2 \cos (2 x)}$

STEP 5

implify the limit by cancelling out the common factor of2.
$\lim{x \rightarrow0} \frac{3 \cos ( x)}{\cos (2 x)}$

STEP 6

Now, we evaluate the limit as x approaches0.
$\frac{3 \cos (6 \cdot0)}{\cos (2 \cdot0)} = \frac{3 \cos (0)}{\cos (0)}$

STEP 7

Since $\cos(0) =1$ , the limit simplifies to $\frac{3 \cdot1}{1} =3$ So, the limit of the function as x approaches0 is3.

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

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches0 for the function $\frac{\sin (6 x)}{\sin ( x)}$ . . We will use the limit properties and trigonometric identities to solve this problem.

STEP 2

STEP 3

STEP 4

Substitute these derivatives back into the limit.
$\lim{x \rightarrow0} \frac{6 \cos (6 x)}{2 \cos (2 x)}$

STEP 5

implify the limit by cancelling out the common factor of2.
$\lim{x \rightarrow0} \frac{3 \cos ( x)}{\cos (2 x)}$

STEP 6

Now, we evaluate the limit as x approaches0.
$\frac{3 \cos (6 \cdot0)}{\cos (2 \cdot0)} = \frac{3 \cos (0)}{\cos (0)}$

STEP 7

Since $\cos(0) =1$ , the limit simplifies to $\frac{3 \cdot1}{1} =3$ So, the limit of the function as x approaches0 is3.

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QuestionFind the limit: lim⁡x→0sin⁡(6x)sin⁡(2x)\lim _{x \rightarrow 0} \frac{\sin (6 x)}{\sin (2 x)}limx→0​sin(2x)sin(6x)​.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches0 for the function sin⁡(6x)sin⁡(x)\frac{\sin (6 x)}{\sin ( x)}sin(x)sin(6x)​. . We will use the limit properties and trigonometric identities to solve this problem.

STEP 2

STEP 3

STEP 4

Substitute these derivatives back into the limit.lim⁡x→06cos⁡(6x)2cos⁡(2x)\lim{x \rightarrow0} \frac{6 \cos (6 x)}{2 \cos (2 x)}limx→02cos(2x)6cos(6x)​

STEP 5

implify the limit by cancelling out the common factor of2.lim⁡x→03cos⁡(x)cos⁡(2x)\lim{x \rightarrow0} \frac{3 \cos ( x)}{\cos (2 x)}limx→0cos(2x)3cos(x)​

STEP 6

Now, we evaluate the limit as x approaches0.3cos⁡(6⋅0)cos⁡(2⋅0)=3cos⁡(0)cos⁡(0)\frac{3 \cos (6 \cdot0)}{\cos (2 \cdot0)} = \frac{3 \cos (0)}{\cos (0)}cos(2⋅0)3cos(6⋅0)​=cos(0)3cos(0)​

STEP 7

Since cos⁡(0)=1\cos(0) =1cos(0)=1, the limit simplifies to3⋅11=3\frac{3 \cdot1}{1} =313⋅1​=3So, the limit of the function as x approaches0 is3.

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

Assumptions1. We are asked to find the limit as x approaches0 for the function $\frac{\sin (6 x)}{\sin ( x)}$ . . We will use the limit properties and trigonometric identities to solve this problem.

Substitute these derivatives back into the limit.
$\lim{x \rightarrow0} \frac{6 \cos (6 x)}{2 \cos (2 x)}$

implify the limit by cancelling out the common factor of2.
$\lim{x \rightarrow0} \frac{3 \cos ( x)}{\cos (2 x)}$

Now, we evaluate the limit as x approaches0.
$\frac{3 \cos (6 \cdot0)}{\cos (2 \cdot0)} = \frac{3 \cos (0)}{\cos (0)}$

Since $\cos(0) =1$ , the limit simplifies to $\frac{3 \cdot1}{1} =3$ So, the limit of the function as x approaches0 is3.