# Trig Limits.

#### spoc21

Hi, I'm having a lot of trouble in solving the following limit problems (all of the, as x --> 0):

1) $$\displaystyle \frac{sin 7x}{sin 4x}$$

2) $$\displaystyle \frac{sin(cos x)}{sec x}$$

[FONT=&quot]I'm very confused with these two problems, and would greatly appreciate any help (I'm not allowed to differentiate the functions).[/FONT]

[FONT=&quot]Thanks [/FONT]

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#### Prove It

MHF Helper
Hi, I'm having a lot of trouble in solving the following limit problems (all of the, as x --> 0):

1) $$\displaystyle \frac{sin 7x}{sin 4x}$$

2) $$\displaystyle \frac{sin(cos x)}{sec x}$$

[FONT=&quot]I'm very confused with these two problems, and would greatly appreciate any help (I'm not allowed to differentiate the functions).[/FONT]

[FONT=&quot]Thanks [/FONT]
$$\displaystyle \lim_{x \to 0}\frac{\sin{7x}}{\sin{4x}} = \lim_{x \to 0}\frac{\sin{(3x + 4x)}}{\sin{4x}}$$

$$\displaystyle = \lim_{x \to 0}\frac{\sin{3x}\cos{4x} + \cos{3x}\sin{4x}}{\sin{4x}}$$

$$\displaystyle = \lim_{x \to 0}\left(\frac{\sin{3x}\cos{4x}}{\sin{4x}} + \cos{3x}\right)$$

$$\displaystyle = \lim_{x \to 0}\left(\frac{\sin{3x}\cos{4x}}{\sin{4x}}\right) + \lim_{x \to 0}\,(\cos{3x})$$

$$\displaystyle = 1 + \lim_{x \to 0}\left(\frac{\sin{3x}\cos{4x}}{\sin{4x}}\right)$$

$$\displaystyle = 1 + \lim_{x \to 0}\left[\frac{(3\sin{x} - 4\sin^3{x})(1 - 2\sin^2{2x})}{2\sin{2x}\cos{2x}}\right]$$

$$\displaystyle = 1 + \lim_{x \to 0}\left[\frac{\sin{x}(3 - 4\sin^2{x})(1 - 2\sin^2{2x})}{4\sin{x}\cos{x}\cos{2x}}\right]$$

$$\displaystyle = 1 + \lim_{x \to 0}\left[\frac{(3 - 4\sin^2{x})(1 - 2\sin^2{2x})}{4\cos{x}\cos{2x}}\right]$$

$$\displaystyle = 1 + \frac{3}{4}$$

$$\displaystyle = \frac{7}{4}$$.

• spoc21 and bigwave

#### Prove It

MHF Helper
Hi, I'm having a lot of trouble in solving the following limit problems (all of the, as x --> 0):

1) $$\displaystyle \frac{sin 7x}{sin 4x}$$

2) $$\displaystyle \frac{sin(cos x)}{sec x}$$

[FONT=&quot]I'm very confused with these two problems, and would greatly appreciate any help (I'm not allowed to differentiate the functions).[/FONT]

[FONT=&quot]Thanks [/FONT]
$$\displaystyle \lim_{x \to 0}\frac{\sin{(\cos{x})}}{\sec{x}} = \lim_{x \to 0}\,[\cos{x}\sin{(\cos{x})}]$$

$$\displaystyle = \cos{(0)}\sin{[\cos{(0)}]}$$

$$\displaystyle = 1\cdot \sin{(1)}$$

$$\displaystyle = \sin{(1)}$$.

• spoc21 and bigwave