The following functions have no Maclaurin expansions since they are not defined at x = 0 except;Question 5Select one:log𝑒⁡𝑥Sinhx𝑒-𝑥𝑥Cosx

To determine which of the functions has a Maclaurin expansion, we need to identify which functions are defined at $x = 0$. A Maclaurin series is a Taylor series expansion of a function around 0.

1. Function Analysis:

   • $\log_e x$: This function is not defined at $x = 0$ because logarithm is undefined for non-positive numbers.
   • $\sinh x$: The hyperbolic sine function is defined at $x = 0$. Specifically, $\sinh(0) = 0$.
   • $e^{-x}$: This function is also defined at $x = 0$. Specifically, $e^{0} = 1$.
   • $\cos x$: The cosine function is defined at $x = 0$. Specifically, $\cos(0) = 1$.

2. Conclusion: The functions $\sinh x$, $e^{-x}$, and $\cos x$ are defined at $x = 0$ and can have Maclaurin expansions. Only $\log_e x$ does not qualify.

Final Answer

The function that has a Maclaurin expansion is: $\sinh x$, $e^{-x}$, and $\cos x$ are defined at $x = 0$; thus, they can have Maclaurin expansions.