Question

Show that f(x) is continuous at x=0:--

Answer 1

f(x)={xsin*1/x; x\[\neq\]0}
& 0 for x=0.

Answer 2

\[ \lim_{x \rightarrow 0} x \sin(1/x) = 0 * some \; value \; between \; -1\; and\; +1 = 0\]

Answer 3

k!

Answer 4

Let \(\delta=\varepsilon\). Then\[|x-0|=|x|<\delta,\]\[|f(x)-f(0)|=|f(x)|=\left|x\sin\left(\frac{1}{x}\right)\right|=|x|\left|\sin\left(\frac{1}{x}\right)\right|\leqslant|x|<\delta=\varepsilon.\]Hence, \(f\) is continuous at \(0\).

Answer 5

please tell me about writing left hand limit & right hand limit for f(x) given

Answer 6

x->a+ means it approaches from right side, in our case, x>0
x->a- means it approaches from right side, if your case, x<0

Either cases, our limit would be zero ..

Answer 7

Thanx to all!