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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 \(0 < |x - 2| < \delta_2\), \(f(x) > k\).

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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.

根据函数极限定义证明以下问题：
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 
已知 \lim_{x \to 2} f(x) = 1，证明：
1. 存在 \delta_1 > 0, 使得当 0 < |x - 2| < \delta_1 时 f(x) > \frac{5}{6}。
2. 对任意 k \in (0, 1)，存在 \delta_2 > 0, 使得当 0 < |x - 2| < \delta_2 时 f(x) > k。

Explore how to prove limits such as \lim_{x \to \infty} \left( 1 + \frac{1}{x^2} \right)^{x^2} = e and \lim_{x \to 1} \frac{3x + 1}{x^2 - 1} = 4 using epsilon-delta definitions. This problem set involves understanding continuity, limit laws, and how to apply the epsilon-delta condition for precise proofs.

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