What is the Limit of the Tangent Function as x Approaches 0?

[erik05]

Hello all. I missed a class in calculus so I didn't get the notes on this so if anyone could explain this question for me, it would be much appreciated.

$$\lim_{x \rightarrow 0} \frac {tanx}{4x}$$
$$= \frac {sinx}{cos4x} ?$$

Not really too sure if I manipulated the equation right. Any hints for the next step? Thanks.

 

[Gmaximus]

You can't do that!



Use l'hopitals rule.

$$\lim_{x \rightarrow 0} \frac {tanx}{4x} =\lim_{x \rightarrow 0} \frac {secxtanx}{4}$$

 

[erik05]

Sorry, I haven't learned l'hopitals rule yet and we're not suppose to use it for these questions.

 

[vincentchan]

expand tanx in taylor series, and do what you should do...

 

[erik05]

This is going to sound really pathetic but no, we haven't the taylor series either.

 

[vincentchan]

do you know the fact that
$$\lim_{x \rightarrow 0} \frac {sinx}{x} =1$$
if yes, you should start from here

 

[dextercioby]

I think Taylor series is taught way after l'Hôpital's rule,don't u think so?

Daniel.

 

[erik05]

do you know the fact that
$$\lim_{x \rightarrow 0} \frac {sinx}{x} =1$$
if yes, you should start from here

That I do know.

 

[dextercioby]

What about "tangent's" definition...?And the limit of cosine as its argument goes to 0 ?

Daniel.

 

[leon1127]

Hello all. I missed a class in calculus so I didn't get the notes on this so if anyone could explain this question for me, it would be much appreciated.

$$\lim_{x \rightarrow 0} \frac {tanx}{4x}$$
$$= \frac {sinx}{cos4x} ?$$

Not really too sure if I manipulated the equation right. Any hints for the next step? Thanks.

= 1/4*(sin[x]/x)*(sec[x])

 

[erik05]

Ah...I got it. Thanks all.

 

[Orion1]

Limit Laminate...


Solution:
$$\boxed{\lim_{x \rightarrow 0} \frac {\tan x}{4x} = \frac{1}{4}}$$

 

[dextercioby]

How fancy that \boxed{...},too bad u don't know "\tan"...

Daniel.

P.S.BTW,I've searched Mathworld and A & S,couln't find this $tanx$ function...

P.P.S.Neither $sinx$,nor $secxtanx$,but i found $\mbox{sinc}\ x$...

P.P.P.S.You edited...