Definition of Function Limit in Mathematics

Learn about the Limit of a Function, one of the fundamental concepts in calculus (differential and integral) and high school mathematics.

📘 What is a Function Limit in Mathematics?

A function limit is one of the basic concepts in calculus and high school mathematics that describes the value a function approaches as the variable gets close to a certain point. Limits are a crucial foundation for learning derivatives (differential) and integrals.

📐 Writing the Limit Notation

In mathematics, a limit is written as:

$$\lim_{x \to a} f(x) = L$$

Meaning:

As $x$ approaches $a$, the function $f(x)$ approaches the value $L$.

A limit can exist even if the function is not defined at that point. This is what distinguishes the value of a limit from the value of the function.

🔄 Simple Example of a Function Limit

For example, the function:

$$f(x) = x + 2$$

Then:

$$\lim_{x \to 3} f(x) = \lim_{x \to 3} (x + 2) = 5$$

Explanation: As $x$ approaches 3, the function value approaches 5. This is the simplest example of a linear function limit.

📊 Approaching Limits Using a Table

If you want to understand limits numerically, you can look at them in the form of an approach table:

$$\lim_{x \to 2} x^2$$

It can be seen that:

$$\lim_{x \to 2} x^2 = 4$$

🧭 Left-Hand and Right-Hand Limits

In calculus, we know:

• Left-hand limit:

  $$\lim_{x \to a^-} f(x)$$

• Right-hand limit:

  $$\lim_{x \to a^+} f(x)$$

🔁 If Both Are Equal:

$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \Rightarrow \lim_{x \to a} f(x) = L$$

❌ Example of a Nonexistent Limit

Consider the following function:

$$f(x) = \begin{cases} 1, & x < 0 \\ 2, & x \geq 0 \end{cases}$$

• $\lim_{x \to 0^-} f(x) = 1$
• $\lim_{x \to 0^+} f(x) = 2$

Because the left and right limits are different, then:

$$\lim_{x \to 0} f(x) \text{ does not exist (limit does not exist)}$$

📝 Example Problems on Function Limits and Their Solutions

Problem 1:

$$\lim_{x \to 3} (2x - 5) = ?$$

Solution:

$$2(3) - 5 = 1$$

Problem 2:

$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} \Rightarrow \frac{(x - 1)(x + 1)}{x - 1} = x + 1 \Rightarrow 2$$

Problem 3:

$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} \Rightarrow \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \Rightarrow 4$$

📌 Conclusion: What is a Function Limit?

• A limit shows the direction a function approaches, not the function’s value at that point.
• A limit can still be calculated even if the function is not defined at that point.
• Understanding limits is very important as a foundation for derivatives and integrals in grades 11 and 12 of high school.

Whats next?????

Properties of Function Limits and Example Problems↝