I concur. This is the result I found, but I wasn't entirely sure since the problem wanted a limit inferior, while obviously you could get a limit. Unfortunately this book is plagued with errors, so I wouldn't be surprised if this is yet another one.

On the function comparison side, a quick plot on Maxima confirms the inequality. Alternatively, you can get the derivative of the function:

and check that the function, starting from 1, decreases close to zero and increases towards infinity to 1 again. So the function stays below 1, although this is not a full proof.

Many Thanks.