Trigonometric limit problem

Member
Hello math helpers and others. Before I ask my question I would like to say that I appreciate everyone's help on these boards, and I hope that I will not be too large of a nuisance in the future. I am in my first calculus class, and it appears that I am going to need a lot of help. I was a member of the previous forum, but I didn't post often, so I hope there isn't any type of limit.

Here's my question.

$$\displaystyle\lim_{\theta\rightarrow 0} {\frac{\sec(\theta)-1}{\theta}}/$$

Um, really don't know where to go from here. Should I expand that numerator to be (1/cos(Θ)) - 1? I know the basic rules of sinx/x = 1, and cosx-1/x = 0, and how to simplify simple things like sin2x/x, but I can't seem to get a start on this one.

Any help would be greatly appreciated. Also, I usually prefer to use Latex, but my typical [itex] tags aren't working, and  centers it. Any advice on getting an in line equation, so I don't just have to type sin(x)/x?

Thanks again,
Mac

*EDIT* I think I figured out the LaTex

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Alexmahone

Active member
Should I expand that numerator to be (1/cos(Θ)) - 1?
Yes. And then use the identity $\displaystyle 1-\cos\theta=2\sin^2\frac{\theta}{2}$.

Member
I'm sorry, I am probably just a bit tired, but I don't understand how you go from $$\frac{\frac{1}{cos{\theta}}-1}{\theta}$$ to $$(1-\cos{\theta})$$ identity. I'll look at this more tomorrow when I am a bit clearer, but if you could elaborate some I would appreciate it.

Thanks again,
Mac

Prove It

Well-known member
MHB Math Helper
I'm sorry, I am probably just a bit tired, but I don't understand how you go from $$\frac{\frac{1}{cos{\theta}}-1}{\theta}$$ to $$(1-\cos{\theta})$$ identity. I'll look at this more tomorrow when I am a bit clearer, but if you could elaborate some I would appreciate it.

Thanks again,
Mac
\displaystyle \begin{align*} \frac{\frac{1}{\cos{\theta}} - 1}{\theta} &= \frac{\frac{1 - \cos{\theta}}{\cos{\theta}}}{\theta} \\ &= \frac{1 - \cos{\theta}}{\theta\cos{\theta}} \\ &= \frac{1 - \cos{\theta}}{\theta} \cdot \frac{1}{\cos{\theta}} \end{align*}

I'm sure you know that the limit of a product is equal to the product of the limits...

Member
Okay, thanks guys. I am following you now. The step I was missing was multiplying all by cos(Θ). It simplifies to 0 * 1 in the bottom.

Very much appreciated, thank you.

CaptainBlack

Well-known member
I'm sure you know that the limit of a product is equal to the product of the limits...
As long as both exist.

$\lim_{x \to 0}\left( x \times (1/x)\right) =1 \ne \left(\lim_{x \to 0} (x) \right) \left(\lim_{x \to 0} (1/x) \right) \text{ which is undefined }$

CB

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