Existence of a limit at a jump point of a ceiling function

For $x$ slightly less than $1.5$, say $x = 1.49$, we get $2x = 2.98$, so $\lceil 2x \rceil = 3$. As $x$ approaches $1.5$ from the left, $2x$ approaches $3$ from below, and the ceiling of any number strictly between $2$ and $3$ is $3$. So $\lim_{x \to 1.5^-} a(x) = 3$.

For $x$ slightly greater than $1.5$, say $x = 1.51$, we get $2x = 3.02$, so $\lceil 2x \rceil = 4$. As $x$ approaches $1.5$ from the right, $2x$ approaches $3$ from above, and the ceiling of any number strictly between $3$ and $4$ is $4$. So $\lim_{x \to 1.5^+} a(x) = 4$. Since the left-hand limit ($3$) and right-hand limit ($4$) are not equal, the two-sided limit $\lim_{x \to 1.5} a(x)$ does not exist.