On a recent first-semester calculus exam, I gave a bunch of limits. The student was supposed to use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other method. One of the problems was this:
Roughly 90% of them asserted that the limit was equal to 1, usually with some variation on the argument that the $+1$ would become negligible compared to $x$ as $x$ got big. Only one student correctly found the limit to be $e$.
I was surprised by the near-universal repetition of the same mistake. I would be interested for others' insights on why this would happen. I suppose a contributing factor might be lack of facility with manipulating exponents. It would have seemed obvious to me (once I'd realized that L'Hospital wouldn't help) to simplify the expression inside the limit, and it also seems obvious to me that exponentials turn addition into multiplication. Probably they are also forming some kind of false analogy with limits of expressions such as $(x+1)/x$. One or two seemed to be using bogus symbolic reasoning with the $\infty$ symbol, simplifying the expression to $e^\infty/e^\infty$.
It also surprised me that so many students made mistakes on these that they could have easily caught simply by plugging in some sample numbers on a calculator, such as $e^9/e^8$ in this example. Maybe it would be helpful to assign homework in the future on which they were required to do this kind of check.