Welcome to our community

Be a part of something great, join today!

Continuous function?

Cbarker1

Active member
Jan 8, 2013
241
Let $f=tan(2x)/x$, x is not equal to 0.

Can the f be defined at x=0 such that it is continuous?


I answered yes. I am wondering if the answer is correct.


Thank you for your help

CBarker1
 

ThePerfectHacker

Well-known member
Jan 26, 2012
236
Compute limit at $0$. What do you get?
 

Cbarker1

Active member
Jan 8, 2013
241
I got 2.
 

ThePerfectHacker

Well-known member
Jan 26, 2012
236
So,
$$\lim_{x\to 0} \frac{\tan 2x}{x} = 2$$
Now define the function,
$$ f(x) = \left\{ \begin{array}{ccc}(\tan x)/x & \text{if} & x\not = 0 \\ 2 & \text{if}& x=0 \end{array} \right. $$

This function is continous everywhere because at $0$ we have $\lim_{x\to 0}f(x) = f(0) = 2$.