Trig Limits.

May 2010
36
0
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
Aug 2008
12,883
4,999
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]
\(\displaystyle \lim_{x \to 0}\frac{\sin{7x}}{\sin{4x}} = \lim_{x \to 0}\frac{\sin{(3x + 4x)}}{\sin{4x}}\)

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

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

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

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

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

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

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

\(\displaystyle = 1 + \frac{3}{4}\)

\(\displaystyle = \frac{7}{4}\).
 
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Prove It

MHF Helper
Aug 2008
12,883
4,999
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]
\(\displaystyle \lim_{x \to 0}\frac{\sin{(\cos{x})}}{\sec{x}} = \lim_{x \to 0}\,[\cos{x}\sin{(\cos{x})}]\)

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

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

\(\displaystyle = \sin{(1)}\).
 
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Reactions: spoc21 and bigwave