The function \( f(x) = \frac{\sin x}{x} \) is defined as \( f(0) = 0 \) for \( x = 0 \) and \( f(x) = \frac{\sin x}{x} \) for \( x \neq 0 \). The limit as \( x \) approaches 0 is \( \lim_{x \to 0} f(x) = 1 \). Since \( f(0) \) does not equal the limit, the discontinuity at \( x = 0 \) is classified as a removable discontinuity, not a jump discontinuity.
PREREQUISITESStudents and educators in calculus, mathematicians analyzing function behavior, and anyone interested in understanding types of discontinuities in mathematical functions.
dwsmith said:$f(x) = \frac{\sin x}{x}$ if $x\neq 0$, $f(0) = 0$.We know that the $\lim\limits_{n\to 0}f(x) = 1$. Since $f(0) = 0$, we will have a jump discontinuity.
Is this correct reasoning?