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For instance, I have signals such as

$$x(t)=e^{\cos(t)}$$

$$x(t)=t\cdot e^{\cos(t)}$$

How should I approach these kinds of signals to determine their fundamental periods?

The fundamental period is the smallest $p > 0$ for which $x(t) = x(t+p)$ holds.

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    Can you add the definition you are using for "fundamental period" to the question? – 2017-01-28
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    Of course, I edited it – 2017-01-28
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    The fundamental period of $x$ is $p$ such $x(t)=x(t+p)$ where p is minimal. So for the first function $2 \pi$ & the second function is not periodic. – 2017-01-28

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For the fist $x(t)$, know that the exponential is a bijective function on the real line, so the period of $x(t)$ is the same as the period of $\cos(t)$, namely $2\pi$. For the second $x(t)$, there is no fundamental period because the function is not periodic. Any continuous periodic function must be bounded, but clearly the second choice of $x(t)$ is not.

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    Are you saying that $2\pi$ is not a real number? Because it is. – 2017-01-28
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    @Paul, I never claimed that $2\pi$ is not a real number. I claimed that the function $x(t) = t e^{\cos(t)}$ is not periodic. You can check that $x(t + 2\pi) = (t+2\pi)e^{\cos(t)} = x(t) + 2\pi e^{\cos(t)} \neq x(t)$. – 2017-01-29
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    What I said was a reply to a comment that is now deleted – 2017-01-29