Limit theorems

Limit Theorems for Students A limit theorem is a powerful tool in studying the behavior of functions as they approach a specific value. These theorems he...

Limit Theorems for Students#

A limit theorem is a powerful tool in studying the behavior of functions as they approach a specific value. These theorems help us understand whether the function approaches a finite value, infinity, or oscillates as the input approaches that value.

Sequential Criterion:

A function $f(x)$ satisfies the sequential criterion if the following two conditions are satisfied:

**For any sequence $\{x_n\}$ converging to $a$ , we have $\lim_{x \to a} f(x) = f(a)$
**If $\{a_n\}$ is increasing and $\{a_n \to a\}$ , then $\lim_{x \to a} f(x) = f(a)$$

Examples:

If $\lim_{x \to 0} f(x) = 1$ and $\lim_{x \to 0} g(x) = 2$ then by the sequential criterion, we have $\lim_{x \to 0} (f(x) + g(x)) = 1 + 2 = 3$
If $\lim_{x \to 1} f(x) = \infty$ and $a > 1$ then by the sequential criterion, we have $\lim_{x \to 1} (f(x))^a = \infty$
If $0 \le a \le 1$ and $f(x) = 0$ for all $x \in (0, 1)$ , then by the sequential criterion, we have $\lim_{x \to 0} f(x) = 0$

These are just a few basic examples of limit theorems and the sequential criterion. Studying limit theorems helps us to understand the behavior of functions and how they approach different values as they approach a specific point

Limit Theorems for Students#

Sequential Criterion:

A function

f(x)

satisfies the sequential criterion if the following two conditions are satisfied:

**For any sequence $\{x_n\}$ converging to $a$ , we have $\lim_{x \to a} f(x) = f(a)$

**If $\{a_n\}$ is increasing and $\{a_n \to a\}$ , then $\lim_{x \to a} f(x) = f(a)$$

Examples:

If $\lim_{x \to 0} f(x) = 1$ and $\lim_{x \to 0} g(x) = 2$ then by the sequential criterion, we have $\lim_{x \to 0} (f(x) + g(x)) = 1 + 2 = 3$

If $\lim_{x \to 1} f(x) = \infty$ and $a > 1$ then by the sequential criterion, we have $\lim_{x \to 1} (f(x))^a = \infty$

If $0 \le a \le 1$ and $f(x) = 0$ for all $x \in (0, 1)$ , then by the sequential criterion, we have $\lim_{x \to 0} f(x) = 0$

Limit theorems

Limit Theorems for Students#

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