Computable Functions converging to Uncomputable Functions

A real number is computable if it can be computed to arbitrary precision.
A real-valued function $f : \mathbb{N} \to \mathbb{R}$ is computable if $f(t)$ is computable for every $t \in \mathbb{N}.$

Does there exist a computable function $f : \mathbb{N} \to \mathbb{R}$ such that $\displaystyle \lim_{t \to \infty} f(t)$ exists but is uncomputable?

Inspiration: CS StackExchange