Find k to make a piecewise function continuous at x=0

[huan.conchito]

Calculus homework studying for the test: Find K such that this is continious

$$f(x)=\left\{\begin{array}{cc}k&\mbox{ if }x=0\\2+\frac{Sin(x)}{x} &\mbox{ if }x\neq 0\end{array}\right$$

i have no clue how to find such a k that this is continious

 

[Data]

well, from your question I see no reason why I can't choose any $k$ that I like. Are you sure that's all there is to it?

Perhaps you are trying to make $f(x)$ continuous?

 

[dextercioby]

U didn't say what the problem wa about.I assume it required that the brach function would be continuous on R...Which means point "x=0",too.

What's the condition for an univariable function defined on a domain $\matcal{D}\subseteq \mathbb{R}$ to be continuous in a point $x \in \mathcal{D}$...?


Daniel.

 

[huan.conchito]

Thats all the question says "find k such that this is continious"

 

[dextercioby]

Then the aswer is $k\in \mathbb{R}$,which means an arbitrary #...

Daniel.

 

[Data]

Thats all the question says "find k such that"
blah blah

perhaps you should elaborate on the "blah blah," verbatim if possible. I find it highly doubtful that that is what it was asking for in a calculus course.

 

[xanthym]

f(x) = {k if x=0 if x == 0
{2+(sinx)/x if x =/= 0

i have no clue how to find such a k

Well, IF the objective is to find "k" such that "f(x)" is continuous, what "k" would satisfy that condition??

HINTS #1 & #2:

$$1: \ \ \ \ \lim_{x \longrightarrow 0} \, f(x) \ = \ f(0)$$

$$2: \ \ \ \ \lim_{x \longrightarrow 0} \, \left (2 \ + \ \frac{\sin(x)}{x} \right ) \ = \ \color{red} \mathbf{?}$$


~~

 

[saltydog]

Well first like this:

$$f(x)=\left\{\begin{array}{cc}k,&\mbox{ if }x=0\\2+\frac{Sin(x)}{x}, &\mbox{ if }x\neq 0\end{array}\right$$

Not too hard, check out the syntax in the editor.

Then just do what Xanthym said to assure continuity.

 

[dextercioby]

Maybe the function needs to be differentiable...

Daniel.

 

[huan.conchito]

the problem is exactly like salty dog wrote it.
and it says
"Find K such that this is continious"

 

[huan.conchito]

$$2: \ \ \ \ \lim_{x \longrightarrow 0} \, \left (2 \ + \ \frac{\sin(x)}{x} \right ) \ = \ \color{red} \mathbf{2?}$$

Well, IF the objective is to find "k" such that "f(x)" is continuous, what "k" would satisfy that condition??

HINTS #1 & #2:

$$1: \ \ \ \ \lim_{x \longrightarrow 0} \, f(x) \ = \ f(0)$$

$$2: \ \ \ \ \lim_{x \longrightarrow 0} \, \left (2 \ + \ \frac{\sin(x)}{x} \right ) \ = \ \color{red} \mathbf{?}$$


~~

 

[dextercioby]

Incorrect.Check that "sin" limit again.

Then u can get conclude what K needs to be...

Daniel.

 

[huan.conchito]

the limit = 0, so K must be 0?

 

[Nylex]

the limit = 0, so K must be 0?

lim {x->0} sin x/x = 1.

 

[huan.conchito]

so k must be 3 ?

 

[Data]

Deleting your previous post and then reposting exactly the same thing is a rather cunning way to get an answer

Anyways, yes, that's fine.

 

[asrodan]

Correct. If you are unsure why $$\ \ \ \ \lim_{x \longrightarrow 0} \, \left (2 \ + \ \frac{\sin(x)}{x} \right ) \ = \ \ \mathbf{1}$$ you might have a look at the Taylor expansion of $$sin(x)$$ .

 

[Data]

Correct. If you are unsure why $$\ \ \ \ \lim_{x \longrightarrow 0} \, \left (2 \ + \ \frac{\sin(x)}{x} \right ) \ = \ \ \mathbf{1}$$ you might have a look at the Taylor expansion.

You mean $3$ .

 

[dextercioby]

We have a thread on the "sinc" limit.It has some good posts.

Daniel.