Gilbert Strang's book on differential equations asks the reader to compare how $e^t$ and $e^{it}$ behave in the complex plane as $t \rightarrow \infty$. $e^t$ blows up along the real axis. Strang then says that $e^{it}$ traces a circle in the complex plane since $e^{it} = \cos(t) + i \sin(t)$. This makes sense when I consider Euler's formula by itself, but now, comparing the two functions, I'm confused. Why doesn't $e^{it}$ blow up (i.e. grow larger and larger) along the imaginary axis?
Edit 1: To clarify, $t$ is real.
Edit 2: I thought that blowing up means the function approaches positive or negative infinity. But $|e^{it}|$ is always $\leq 1$.