Question
Let $$f\left( x \right) = \sin \frac{1}{x},\,x \ne 0.$$ Then $$f\left( x \right)$$ can be continuous at $$x=0$$
A.
if $$f\left( 0 \right) = 1$$
B.
if $$f\left( 0 \right) = 0$$
C.
if $$f\left( 0 \right) = - 1$$
D.
for no value of $$f\left( 0 \right)$$
Answer :
for no value of $$f\left( 0 \right)$$
Solution :
For continuity at $$x = 0,\,\mathop {\lim }\limits_{h \to 0} \,\sin \frac{1}{{0 + h}} = \mathop {\lim }\limits_{h \to 0} \,\sin \frac{1}{{0 - h}} = f\left( 0 \right)$$
But $$\mathop {\lim }\limits_{h \to 0} \,\sin \frac{1}{{0 + h}} = \mathop {\lim }\limits_{h \to 0} $$
{ a number between 1 and $$-1$$ } $$ \ne a$$ definite number.
$$\therefore \,f\left( 0 \right)$$ cannot be a definite number.