A Guide to Choosing Criteria

In practice, choosing the right criterion for the existence of a limit is an important question. This guide will help you select the most appropriate criterion based on the specific situation.

Strategies for Choosing a Criterion

When to Use the Squeeze Theorem

Applicable situations:

• The function is “sandwiched” between other functions
• The limit is difficult to compute directly
• The function has an oscillatory nature
• It involves the boundedness of trigonometric functions

Typical examples:

• $\lim_{x \to 0} x \sin \frac{1}{x}$
• $\lim_{x \to 0} x^2 \cos \frac{1}{x}$
• $\lim_{x \to \infty} \frac{\sin x}{x}$

When to Use the Monotone Convergence Criterion

Applicable situations:

• The sequence has obvious monotonicity
• The sequence is bounded
• The limit cannot be found directly
• The sequence is defined by a recurrence relation

Typical examples:

• $x_n = \left(1 + \frac{1}{n}\right)^n$
• $x_n = \frac{n}{n+1}$
• $x_n = \sqrt[n]{n}$

When to Use the Cauchy Criterion

Applicable situations:

• The monotonicity of the sequence is not obvious
• Convergence must be proved without being able to find the limit
• The sequence has a complicated recurrence relation
• The sequence involves irrational or transcendental numbers

Typical examples:

• $x_n = \sum_{k=1}^{n} \frac{1}{k^2}$
• $x_n = \sum_{k=1}^{n} \frac{1}{k!}$

When to Use the Subsequence Criterion

Applicable situations:

• Divergence must be proved
• The sequence has obvious periodicity
• Subsequences converging to different limits can be found

Typical examples:

• $x_n = (-1)^n$
• $x_n = \sin \frac{n\pi}{2}$

When to Use the Boundedness Criterion

Applicable situations:

• Divergence must be proved
• The sequence is obviously unbounded
• The sequence tends to infinity

Typical examples:

• $x_n = n$
• $x_n = n^2$
• $x_n = 2^n$

Comprehensive Application

Strategies for Solving Difficult Problems

1. First analyze the properties of the sequence/function

   • Is there obvious monotonicity?
   • Is it bounded?
   • Is there periodicity?

2. Try to compute the limit directly

   • If possible, compute it directly
   • If difficult, consider using a criterion

3. Choose an appropriate criterion

   • Choose based on the features of the sequence/function
   • Several criteria may need to be combined

4. Verify the result

   • Check whether the computation is correct
   • Verify whether the conclusion is reasonable

Practice Problems

Exercise 1

For the sequence $x_n = \frac{n^2 + 1}{n^2 + n}$, which criterion should be used? Why?

Exercise 2

For the limit $\lim_{x \to 0} x \sin \frac{1}{x}$, which criterion should be used? Why?

Exercise 3

For the sequence $x_n = \sum_{k=1}^{n} \frac{1}{k^2}$, which criterion should be used? Why?

Exercise 4

For the sequence $x_n = (-1)^n$, which criterion should be used? Why?

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Summary

Symbols Used in This Article

中英对照