How to Simplify a Trigonometric Limit and Solve for 1/2

[MassInertia]

This is from Thomas' Calculus Early Trancendentals, Media Upgrade, page 108, #27:

The limit as x approaches 0 of (x csc(2x))/cos(5x)

The answer is 1/2, which graphing confirms, but hell if I know how to get rid of the 5x and still come up with 1/2. It is boggling my mind. Any help would be appreciated.

 

[rock.freak667]

csc(2x)=1/sin(2x)if I remember correctly

$$\lim_{x \rightarrow 0} \frac{x}{sinx} =1$$

 

[MassInertia]

Yes, that much I know. The problem I have is with the cos(5x). How does one deal with that and come up with 1/2 as the final answer?

 

[Gregg]

Re-arrange then use L'Hopital

$$\displaystyle\lim_{x\to 0} \frac{x\csc(2x)}{\cos(5x)}$$

$$= \displaystyle\lim_{x\to 0} \frac{x\csc(2x)}{1}$$

$$= \displaystyle\lim_{x\to 0} \frac{x}{\sin(2x)} \to \frac{0}{0}$$

$$= \displaystyle\lim_{x\to 0} \frac{1}{2\cos(2x)} = \frac{1}{2}$$

 

[MassInertia]

Well, we haven't covered L'Hopital yet. Also, the denominator is cos(5x), not 5x.

 

[Bohrok]

You need to rewrite the limit to use other limits that you know, such as sinx/x

$$\lim_{x \rightarrow 0} \frac{x\csc2x}{\cos5x} = \lim_{x \rightarrow 0} (x\csc2x * \frac{1}{\cos5x}) = \lim_{x \rightarrow 0} \frac{x}{\sin2x} * \lim_{x \rightarrow 0} \frac{1}{\cos5x}$$

$$= \lim_{x \rightarrow 0} \frac{1}{2} \frac{2x}{\sin2x} * \lim_{x \rightarrow 0} \frac{1}{\cos5x} = \frac{1}{2} \lim_{x \rightarrow 0} \frac{1}{\frac{\sin2x}{2x}} * \lim_{x \rightarrow 0} \frac{1}{\cos5x}$$

For $$\lim_{x \rightarrow 0} \frac{sin2x}{2x}$$

let k = 2x. Since 2x goes to 0 as k goes to 0,

$$\lim_{x \rightarrow 0} \frac{\sin2x}{2x} = \lim_{k \rightarrow 0} \frac{\sin k}{k}$$

 

[Gregg]

I made a typo in latex, if it were 5x then we'd be in trouble.

L'Hopital

$$\displaystyle\lim_{x\to a} \frac{f(x)}{g(x)} = \displaystyle\lim_{x\to a}\frac{f'(x)}{g'(x)}$$

if f(a)=g(a)=0 (or infinity)

$$\frac{f(x)}{g(x)}=\frac{f(a)+(x-a)f'(a)+\frac{1}{2!}(x-a)^2g''(a)\cdots}{g(a)+(x-a)g'(a)+\frac{1}{2!}(x-a)^2g''(a)+\cdots}=\frac{(x-a)f'(a)+\frac{1}{2!}(x-a)^2g''(a)\cdots}{(x-a)g'(a)+\frac{1}{2!}(x-a)^2g''(a)+\cdots}=\frac{f'(a)+(x-a)g''(a)\cdots}{g'(a)+(x-a)g''(a)+\cdots}=\frac{f'(x)}{g'(x)}$$

f(a) and g(a) disappear and can divide top and bottom by 1/2(x-a).

example

$$\displaystyle\lim_{x\to 0} \frac{\sin (x)}{x}$$

Top and bottom tend to 0, so,

$$\displaystyle\lim_{x\to 0} \frac{\sin (x)}{x} =\displaystyle\lim_{x\to 0} \frac{\cos (x)}{1} = 1$$

 

[Bohrok]

I've read in my calc book and elsewhere that you shouldn't use L'Hopital's rule with sinx/x since that limit is used in proofs to find the derivative of sine. Circular reasoning...

 

[MassInertia]

Thanks, Bohrok, that's exactly what I needed to set my brain straight.

Perhaps if I would have remembered that the cosine of zero is one, I wouldn't have been in such a pickle. Hah.

Thanks to everyone for their assistance.