Alice, Bob, Cecilia and Dan were tasked with finding the limit of the function $f$ given by its graph (see the figure below) at the point $x = 3$.
From the graph, Alice realized that $f(3) = 6$, and therefore concluded that the limit of a function $f$ at the point $x = 3$ exists and equals $6$. She wrote: $$ \lim_{x \rightarrow 3} f(x) = 6 $$
Bob claimed that the function is not continuous at the point $x = 3$. Therefore, he argued that $f$ has no limit at that point.
Cecilia expressed and evaluated the one-sided limits, and compared them with the function value at $x=3$: $$ \begin{gather} \lim_{x \rightarrow 3^+} f(x) = 4 \cr \lim_{x \rightarrow 3^-} f(x) = 4 \cr f(3) = 6 \end{gather} $$ Since the function value $f(3)$ differs from the one-sided limits, she concluded that the limit does not exist.
Dan claimed that both one-sided limits equal $4$. Therefore, he stated that the correct answer is: $$ \lim_{x \rightarrow 3} f(x) = 4. $$ Who was right?
Dan
Cecilia
Bob
Alice
None of them
The value of our function is equal to $6$ at $x = 3$. However. the limit at this point is $4$ because both the right and the left limits equal $4$ at $x = 3$.
