MHB Is the Function f=tan(2x)/x Continuous at x=0?

[cbarker1]

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]

Compute limit at $0$. What do you get?

 

[cbarker1]

I got 2.

 

[ThePerfectHacker]

I got 2.

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 continuous everywhere because at $0$ we have $\lim_{x\to 0}f(x) = f(0) = 2$.