Let f(x)=x \sin \frac{1}{x} for x0 and g(x)=x^2 \sin \frac{1}{x} for x0 . If possible define f(0),g(0) so that f,g are right continuous at 0 . Which of the following statements are True?

Question

Let   f(x)=x \sin \frac{1}{x}  for   x>0  and   g(x)=x^2 \sin \frac{1}{x}  for   x>0 . If possible define    f(0),g(0)  so that   f,g  are right continuous at   0 . Which of the following statements are True?

Accepted Answer

g  is differentiable on   [0,1] .

Accepted Answer

\lim_{x	o 0^+}\frac{f(x)}{g(x)}=\infty

Accepted Answer

\lim_{x	o 0^+}\frac{g(x)}{f(x)}=0

Accepted Answer

f,g  are continuous on   [0,1] .

Accepted Answer

f,g  are differentiable on   (0,1) .

Answer

L'Hopital rule can be applied to find   \lim_{x	o 0^+}\frac{f(x)}{g(x)}

Answer

L'Hopital rule can be applied to find   \lim_{x	o 0^+}\frac{g(x)}{f(x)}

Answer

f'(x)
eq 0  on   (0,1) .

Answer

f  is differentiable on   [0,1] .

Answer

\lim_{x	o 0^+}\frac{f'(x)}{g'(x)}  exists

Answer

\lim_{x	o 0^+}\frac{g'(x)}{f'(x)}  exists

Answer

g'(x)
eq 0  on   (0,1) .

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Let f(x)=x \sin \frac{1}{x} for x>0 and g(x)=x^2 \sin \frac{1}{x} for ...

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