Understanding the limits of sequences can be tricky. Many students run into problems that might lead them to wrong answers about whether a sequence is going somewhere (converging) or running off without a limit (diverging). Here are some common mistakes to watch out for:

Confusing Convergence Rules
One big mistake is not fully understanding what it means for a sequence to converge. A sequence, like $\{a_n\}$, is said to converge to a number $L$ if, no matter how small you want to get (we call this $\epsilon$), there is a point in the sequence (let's call it $N$) after which all the terms get closer and closer to $L$. Many students forget that convergence is about how the sequence behaves as it goes on forever, not just how it looks at the beginning.

Skipping Divergence Checks
Sometimes, students assume a sequence converges without checking if it actually might be diverging. Some sequences wiggle around or grow endlessly. For example, the sequence $\{(-1)^n\}$ bounces between -1 and 1 and does not settle down. If you think a sequence converges, you should really double-check the definition of convergence.

Ignoring Confusing Forms
Occasionally, you might run into tricky situations where limits seem to turn into things like $\frac{0}{0}$ or $\infty - \infty$. This can lead to mistaken beliefs about whether a sequence converges. When you hit these tricky cases, it's important to use methods like L'Hôpital's Rule or some algebra tricks to sort them out properly.

Relying Too Much on Gut Feelings
Many times, students trust their gut feelings instead of sticking to the math. Just because it looks like a sequence is heading towards a limit in the beginning doesn’t mean it will keep doing so forever. You need to look at the overall pattern and make sure that all the following terms point in the same direction.

Forgetting About Cauchy Sequences
Another common mistake is not checking whether a sequence is a Cauchy sequence. A sequence $\{a_n\}$ is called Cauchy if, for every small distance you choose (again, $\epsilon$), there’s a point $N$ past which all the terms are very close to each other. Recognizing that a sequence can converge if it’s Cauchy is important, especially in cases where it’s not immediately clear.

Not Using Convergence Tests Properly
Sometimes, students don’t make the best use of tests that help determine if sequences converge. Using things like the Monotone Convergence Theorem or the Squeeze Theorem can really help clarify whether a sequence is converging or diverging. If these tests are ignored, students might get a confusing picture of what’s going on.

To effectively analyze the limits of sequences, it’s important to avoid these pitfalls by being systematic and careful. If you stick to the definitions and use the right mathematical tools, you can steer clear of common mistakes that lead to wrong conclusions. Understanding sequences is crucial for calculus and will help prepare you for even more advanced topics down the road.