Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this.
-$$f ( x ) = \left\{ \begin{array} { c c }
( \sin x ) ^ { x } , & x \neq 0 \
c , & x = 0
\end{array} ; \quad x = 0 \right.$$
A) -1
B) 0
C) 1
D) Impossible

Question

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this.
-$$f ( x ) = \left\{ \begin{array} { c c } 
( \sin x ) ^ { x } , & x \neq 0 \
c , & x = 0
\end{array} ; \quad x = 0 \right.$$
A) -1
B) 0
C) 1
D) Impossible

Quizplus · Accepted Answer

To find the value of c that makes the function continuous at x = 0, we need to check if the left-hand limit and the right-hand limit approach the same value as x approaches 0. 
$\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^-} (\sin x)^x = (\sin 0)^0 = 1^0 = 1$
$\lim_{x 	o 0^+} f(x) = \lim_{x 	o 0^+} c = c$
Therefore, for the function to be continuous at x = 0, we need $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x)$, which means c = 1. Hence, the answer is (C).

Find a Value of C That Makes the Function Continuous f(x)={(sin⁡x)x,x≠0c,x=0;x=0f ( x ) = \left\{ \begin{array} { c c } ( \sin x ) ^ { x } , & x \neq 0 \\c , & x = 0\end{array} ; \quad x = 0 \right.f(x)={(sinx)x,c,​x=0x=0​;x=0

Find a Value of C That Makes the Function Continuous $f ( x ) = \left\{ \begin{array} { c c } ( \sin x ) ^ { x } , & x \neq 0 \\c , & x = 0\end{array} ; \quad x = 0 \right.$