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

• erik05
In summary, Daniel didn't understand the question and was having trouble getting an answer. He found the limit of tanx as x approaches 0 on his own and found that it is 1/4.
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.

You can't do that!

Use l'hopitals rule.

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

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

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

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

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

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

Daniel.

vincentchan said:
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.

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

Daniel.

erik05 said:
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])

Ah...I got it. Thanks all.

Limit Laminate...

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

Last edited:
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...

Last edited:

## 1. What are the basic trigonometric functions?

The basic trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. These functions are used to relate the angles and sides of a right triangle.

## 2. What is the domain of trigonometric functions?

The domain of trigonometric functions is all real numbers, as the input (angle) can be any value. However, for certain trigonometric functions, the domain may be restricted to ensure that the function is well-defined.

## 3. What are the limits of trigonometric functions?

The limits of trigonometric functions depend on the value of the input (angle). For example, the limit of sine and cosine functions is between -1 and 1, while the limit of tangent and cotangent functions does not exist at certain values.

## 4. How do you find the limit of a trigonometric function?

To find the limit of a trigonometric function, you can use the properties of limits and trigonometric identities. You may also need to use L'Hopital's rule or graphing techniques to determine the limit at certain values.

## 5. What is the purpose of studying limits of trigonometric functions?

Studying limits of trigonometric functions helps us understand the behavior of these functions as the input (angle) approaches certain values. This knowledge is useful in calculus and other fields of mathematics, as well as in real-life applications such as physics and engineering.

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