Question

Prove sin³x and sinx³ are continuous on R

Answer 1

any function, f(x) is continuous at x = a only when $\displaystyle\lim_{h\to a^+}f(x)=\displaystyle\lim_{h\to a^-}f(x)=f(a)$

here, f(x) = sin³x,
in case of cosine function, f(a+) = f(a-) = f(a)
for understanding let's take x = 0,
at x = 0, f(0^+) = $\displaystyle\lim_{h\to 0^+}sin^3x$=(sin0)³ = 0
f(0^-)=$\displaystyle\lim_{h\to 0^-}sin^3x$ = (sin0)³ = 0
and f(0) = sin³0 = (0)³ = 0
hence , it follows the condition of continuity.
therefore , f(x) = sin³x is continuous at x = 0
similarly, you can choose any point from all real numbers. you will get , f(x) follows the condition.
hence, finally we can say that f(x) = sin³x is continuous on R.


any function, f(x) is continuous at x = a only when $\displaystyle\lim_{h\to a^+}f(x)=\displaystyle\lim_{h\to a^-}f(x)=f(a)$

here, f(x) = sinx³,
in case of cosine function, f(a+) = f(a-) = f(a)
for understanding let's take x = 0,
at x = 0, f(0^+) = $\displaystyle\lim_{h\to 0^+}sinx^3$=(sin0³)= sin0 = 0
f(0^-)=$\displaystyle\lim_{h\to 0^-}sinx^3$ = (sin0³)= sin0 = 0
and f(0) = sin0³ = sin0 = 0
hence , it follows the condition of continuity.
therefore , f(x) = sinx³ is continuous at x = 0
similarly, you can choose any point from all real numbers. you will get , f(x) follows the condition.
hence, finally we can say that f(x) = sinx³ is continuous on R.