Limits with trigonometric functions.

**amillionwinters** · September 16th 2012, 02:17 PM

I am having trouble with this limit. I don't need an answer, but I would appreciate if someone could give me a hint or show me what I am missing.

The following is what I can make sense of it so far:

$\lim_{x \to 0}\frac{tan\theta - sin\theta}{sin^3\theta}$

$\rightarrow \lim_{x \to 0}\frac{\frac{sin\theta}{cos\theta} - sin\theta}{sin^3\theta}$

$\rightarrow \lim_{x \to 0}\left(\frac{sin\theta}{cos\theta}\right)\left( \frac{1}{sin^3\theta}\right) - \frac{sin\theta}{sin^3\theta}$

$\rightarrow \lim_{x \to 0}\frac{1}{cos\theta cos^2\theta} - \frac{1}{sin^2\theta}$

$\rightarrow \lim_{x \to 0}\frac{1- cos\theta}{cos\theta sin^2\theta}$

I am aware that $\lim_{x \to 0}\frac{1- cos x}{x} = 0$ ; however, I am a little lost on how to get there, or if that is even the direction I should be taking. Any hints or explanations would be greatly appreciated. Thanks!

Take care.
/alan

**SworD** · September 16th 2012, 02:29 PM

Factor out a $\sin(\theta)$ from both the numerator and denominator of your original ratio, and see where you can go from there.

**AltF4** · September 16th 2012, 02:34 PM

I believe you will need to apply L'Hôpital's rule.
$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$

**skeeter** · September 16th 2012, 02:47 PM

$\frac{\tan{x} - \sin{x}}{\sin^3{x}} =$

$\frac{\sin{x} - \cos{x}\sin{x}}{\cos{x}\sin^3{x}} =$

$\frac{1 - \cos{x}}{\cos{x}\sin^2{x}} =$

$\frac{1 - \cos{x}}{\cos{x}(1-\cos^2{x})} =$

$\frac{1}{\cos{x}(1+\cos{x})}$

**skeeter** · September 16th 2012, 03:16 PM

Originally Posted by AltF4

I believe you will need to apply L'Hôpital's rule.
$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$

why don't you show the OP that method?

Thread: Limits with trigonometric functions.

LinkBack

Thread Tools

Display

Limits with trigonometric functions.

Re: Limits with trigonometric functions.

Re: Limits with trigonometric functions.

Re: Limits with trigonometric functions.

Re: Limits with trigonometric functions.

Similar Math Help Forum Discussions

Applications of Derivatives of Trigonometric and Inverse Trigonometric Functions

Limits at infinity of Trigonometric Functions

Limits Involving Trigonometric Functions

infinite limits on trigonometric functions

Limits of trigonometric functions

Search Tags