Question

please help. will medal.

Answer 1

prove that \[x \sin (1/x)\] is continuous

Answer 2

at x=/=0 and F(0)=0

Answer 3

you can do this by proving the continuity of x then continuity of sin (1/x)
then the product of the 2 functions in continuous

Answer 4

i know that lim of x as x to 0 is continuous but i can't prove sin1/x. please help me

Answer 5

well you can graph it and see how that turns out
if it goes on and on it is continuous

Answer 6

i prefer the ction not the graph. please if you can, please help working it out

Answer 7

i dont think sin (1/x) is continuous at x = 0 because 1/0 is indeterminate

Answer 8

\[f(x)=\begin{cases}x\sin\dfrac{1}{x}&\text{for }x\neq0\\0&\text{for }x=0\end{cases}\]
To establish continuity (namely, at \(x=0\), you have to show that the following are satisfied:
\[\lim_{x\to0^-}f(x)=\lim_{x\to0^+}f(x)=L\quad\text{and}\quad f(0)=L\]
In this case, you need to show that \(L=0\), since you're given that \(f(0)=0\).

Hint: Replace \(x\) with \(t\) using the substitution \(t=\dfrac{1}{x}\). Then as \(x\to0^-\), you have \(t\to-\infty\).
\[\lim_{x\to0^-}x\sin\frac{1}{x}=\lim_{t\to-\infty}\frac{\sin t}{t}=\cdots\]
You can treat the right-sided limit similarly.

Answer 9

@nincompoop and @dan815

Answer 10

@perl