My course

Hi there! Thank you for reporting the tech issue.

Hi there! With closed circle, the function and limit value are likely to be the same - We may not see why we need to take find limits.

With open circle, the function value is likely undefined/doe not exist but limit may exist, (like when denominator is zero, $\lim_{x\rightarrow1}\left(\frac{1}{x-1}\right)$).

The limits exists when 'right limit = left limit'. The two limits should approach the same value when 'right limit = left limit'.

When defining limits, we don't mind about closed and open circles, we focus on 'right limit = left limit'.

More observations: The absolute value function is undefined if |f (x)| = 0. For example, |x^3 - x| is undefined at x = -1, x = 0, and x = +1. We have open circles at x = -1, x = 0, and x = +1, we need to use limits to find the function value since the function can't be evaluated directly.

The concept of limits appears meaningless until we meet situations where we can't compute f(x).

Why do we need to check if the regular limit is equal to the function of x AFTER we have already found if it is discontinuous or continuous?