Question

consider the function f(x) = {(sinx)/x, x cannot equal 0
{k, , x = 0
In order for f(x) to be continuous at x - 0, the value of k must be...

Answer 1

We have that

\[\lim_{x \rightarrow 0} (\sin x)/x =1\]
(This is a common result; I'm sure you can find a proof of it somewhere on Google if you need to)

Now in order for a function to be continuous at 0, we require that
\[\lim_{x \rightarrow 0} f(x) =f(0)\]
It's pretty clear that the only k for which this is true is k=1.

Answer 2

sorry, just slightly confused on the end about the k

Answer 3

Zid's got it exactly right. In order for a function to be continuous, the value of the function at every point has to be equal to the value of the limit as x approaches that point. Thus, since \[\lim_{x \rightarrow 0} \sin(x)/x = 1\], we pick k to be 1 so that at x=0, f(x) = 1.

Answer 4

ok, that makes sense.