The function signum (sgn) is defined to assign $-1$ to negative numbers, $0$ to zero, and $+1$ to positive numbers. Adam was asked to find the limit of the function $$ f(x) = \mathrm{sgn} (x - 3) +2 $$ at the point $x = 3$. From his math lessons, he remembered the procedure that involves calculating the function value, and the result equals the desired limit $$ \lim_{x\rightarrow 3} \left[\mathrm{sgn} (x - 3) + 2\right] = \mathrm{sgn} (3-3) + 2 = \mathrm{sgn} \,0 + 2 = 0 + 2 = 2. $$
His classmates then commented on his way of solving. Who is right?
Alice: Adam is wrong. The value of the function $\mathrm{sgn} $ at the point $x = 3$ equals $+1$. Adding the number $2$ gives us the sought limit, which equals $3$.
Bob: Adam is wrong. The function $f$ at the point $x = 3$ is not continuous. Therefore, the limit cannot be equal to the function value.
Chris: Adam is correct. The limit can always be calculated as the function value.
David: Adam is wrong. The correct way of solving is: $$ \lim_{x\rightarrow 3} \left[\mathrm{sgn} (x - 3) + 2\right] = \lim_{x\rightarrow 3} \left[\mathrm{sgn} (x - 1) \right]=\mathrm{sgn}\,2 =1. $$
Bob
Alice
Chris
David
The function $f$ is not continuous at the point $x = 3$. Therefore, the limit cannot be equal to the function value. The limit from the left is $1$, and the limit from the right is $3$; the two-sided limit doesn’t exist.