Understanding the Concept of Function Limit

Sahabat Smaridasa - Mathematical Foundation

In everyday life, we often hear words like "almost" or "approaching". For example, "Messi almost scored a goal", "the speed of the motorcycle is approaching 110 km/h", and so on. The words "almost" or "approaching" in mathematics are called limits. In this article, we will study the Definition of Function Limits. The function limits discussed here refer to "algebraic function limits" and "trigonometric function limits" which will be covered in other articles. In mathematics, a limit represents the approximate value of a variable as it approaches a real number. The following is the limit notation.

Definition of Function Limit

Let \( f \) be a function \( f : \mathbb{R} \to \mathbb{R} \) and let \( L \) and \( a \) be real numbers.

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

if and only if \( f(x) \) approaches \( L \) for all \( x \) approaching \( a \).

How to read limit notation: \( \lim_{x \to a} f(x) = L \) is read as "the limit of function \( f(x) \) as \( x \) approaches \( a \) equals \( L \)".

Methods for Evaluating Limits

To determine the limit value of a function, there are several methods:

1. Numerical Method (tabulation)
2. Direct Substitution
3. Factorization
4. Multiplying by the Conjugate
5. Using Derivatives (L'Hôpital's rule)

In this article on the definition of limits, we will only use the numerical method. The numerical method is a way of calculating limits by substituting values from the left and right sides and listing results in a table. However, this method is less efficient because building a table takes time.

📘 Example 1 :

Find the limit of the function \( f(x) = x + 1 \) as \( x \) approaches \( 2 \).

Solution:

1. The problem can be written as: \(\displaystyle \lim_{x \to 2} (x + 1) = \ldots\)
2. Using the numerical method, we choose values of \( x \) approaching 2 from the left and right, then substitute them into \( (x + 1) \). The results are shown in the following table.

Table of values for \( f(x) = x + 1 \) as \( x \) approaches 2

\(x\)	1.9	1.99	1.999	1.9999	→ 2 ←	2.0001	2.001	2.01	2.1
\(f(x)=x+1\)	2.9	2.99	2.999	2.9999	?	3.0001	3.001	3.01	3.1

Gambar 1: Ilustrasi nilai fungsi \( f(x) = x + 1 \) saat x mendekati 2 (pendekatan numerik)

From the table and illustration above, we see that from the left side of 2, the function values approach 2.9999. From the right side of 2, the function values approach 3.0001. This means that the limit of \( f(x) = x + 1 \) as \( x \) approaches 2 is 3. Hence, \(\displaystyle \lim_{x \to 2} (x + 1) = 3\).

Condition for a Function to Have a Limit at a Point

A limit is said to exist if the left-hand limit and the right-hand limit are equal. The left-hand limit is the value the function approaches from the left, denoted \(\displaystyle \lim_{x \to a^-} f(x)\). The right-hand limit is the value the function approaches from the right, denoted \(\displaystyle \lim_{x \to a^+} f(x)\).

That is, if \(\displaystyle \lim_{x \to a^-} f(x) = L\) and \(\displaystyle \lim_{x \to a^+} f(x) = L\), then \(\displaystyle \lim_{x \to a} f(x) = L\).

Description of whether a limit exists for a function \( f(x) \) as \( x \) approaches \( c \).

Gambar 2: Ilustrasi keberadaan limit berdasarkan limit kiri dan kanan (Figure A, B, C, D)

Based on the graphical description above:

• Figure A : The limit exists because the function approaches the same value from both sides (continuous).
• Figure B : The limit does not exist because the left-hand limit ≠ right-hand limit (jump discontinuity).
• Figure C : The limit exists because the left-hand limit = right-hand limit, despite a hole at the point.
• Figure D : The limit does not exist because the left-hand limit ≠ right-hand limit (different asymptotic behavior).

📘 Example 2 :

Does the following function have a limit as \( x \) approaches \( 1 \)?

\[ f(x) = \begin{cases} x^2, & \text{if } x \le 1 \\ x + 1, & \text{if } x > 1 \end{cases} \]

Solution:

• If \( x \le 1 \), then \( f(x) = x^2 \).
• If \( x > 1 \), then \( f(x) = x + 1 \).

Table of values approaching 1 from the left and right:

\(x\)00.50.70.90.990.999→ 1 ←1.0011.011.11.51.7 \(f(x)\)00.250.490.810.98010.998001?2.0012.012.12.52.7

Analysis of left-hand and right-hand limits:
Left-hand limit: from the left approaching 1, the function values approach \(0.998001 \approx 1\), so \(\displaystyle \lim_{x \to 1^-} f(x) = 1\).
Right-hand limit: from the right approaching 1, the function values approach \(2.001 \approx 2\), so \(\displaystyle \lim_{x \to 1^+} f(x) = 2\).
Because the left-hand limit and right-hand limit are not equal, the function \( f(x) = \begin{cases} x^2 & x \le 1 \\ x+1 & x > 1 \end{cases} \) as \( x \) approaches 1 does NOT have a limit.

Gambar 3: Grafik fungsi \( f(x) = \begin{cases} x^2 & x \le 1 \\ x+1 & x > 1 \end{cases} \) di sekitar x = 1 (loncatan)

⚠️ Conclusion: The function \( f(x) = \begin{cases} x^2, & x \le 1 \\ x+1, & x > 1 \end{cases} \) as \( x \) approaches 1 does NOT have a limit.