The image contains problems related to limits in calculus. Here is a translation of the mathematical problems in the image:

Problem 1:

Using the definition of the limit, prove:

1. $\lim_{x \to \infty} \left( 1 + \frac{1}{x^2} \right)^{x^2} = e$
2. $\lim_{x \to 1} \frac{3x + 1}{x^2 - 1} = 4$

Problem 2:

It is known that $\lim_{x \to 2} f(x) = 1$. Prove the following:

1. There exists $\delta_1 > 0$, such that when $0 < |x - 2| < \delta_1$, $f(x) > \frac{5}{6}$.
2. For any constant $k \in (0, 1)$, there exists $\delta_2 > 0$, such that when $0 < |x - 2| < \delta_2$, $f(x) > k$.

If you'd like, I can solve or guide you through any of these problems. Let me know which specific problem you'd like to work on!

Here are 5 related questions you might find helpful:

1. How do you formally define the limit of a function as $x \to a$?
2. What is the general method for proving limits using $\epsilon$-$\delta$ definitions?
3. How can you solve limits that involve indeterminate forms?
4. What strategies can be used to evaluate limits at infinity?
5. How does the continuity of a function affect the existence of limits?

Tip: When working with limits, always check for indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, as these often indicate the need for algebraic manipulation or L'Hopital's Rule.