The concept of a limit describes the behavior of a function as it approaches a certain value. It is a fundamental idea in calculus used to define derivatives and integrals.
1.1 Definition of a Limit
If \( f(x) \) gets arbitrarily close to \( L \) as \( x \) approaches \( a \), then we write:
\[ \lim\limits_{x \to a} f(x) = L \]1.2 One-Sided Limits
Limits can be evaluated from the left or right:
- Left-hand limit: \( \lim\limits_{x \to a^-} f(x) \)
- Right-hand limit: \( \lim\limits_{x \to a^+} f(x) \)
The two-sided limit exists if and only if:
\[ \lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x) \]2. Limit Laws
The following properties help evaluate limits efficiently:
- \( \lim\limits_{x \to a} (f(x) + g(x)) = \lim\limits_{x \to a} f(x) + \lim\limits_{x \to a} g(x) \)
- \( \lim\limits_{x \to a} (cf(x)) = c \lim\limits_{x \to a} f(x) \)
- \( \lim\limits_{x \to a} (f(x)g(x)) = (\lim\limits_{x \to a} f(x)) (\lim\limits_{x \to a} g(x)) \)
- \( \lim\limits_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)} \), provided \( \lim\limits_{x \to a} g(x) \neq 0 \)
Example 1: Evaluating a Basic Limit
Find:
\[ \lim\limits_{x \to 3} (2x^2 - 5x + 1) \]Using direct substitution:
\[ 2(3)^2 - 5(3) + 1 = 18 - 15 + 1 = 4 \]3. Continuity
A function \( f(x) \) is continuous at \( x = a \) if:
- \( \lim\limits_{x \to a} f(x) \) exists
- \( f(a) \) is defined
- \( \lim\limits_{x \to a} f(x) = f(a) \)
Example 2: Checking Continuity
Determine if \( f(x) \) is continuous at \( x = 2 \):
\[ f(x) = \begin{cases} x^2 - 3, & x < 2 \\ 4, & x = 2 \\ 2x + 1, & x > 2 \end{cases} \]Practice Problems
Problem 1: Compute:
\[ \lim\limits_{x \to 2} (x^3 - 4x + 2) \]Solution:
Using direct substitution:
\[ (2)^3 - 4(2) + 2 = 8 - 8 + 2 = 2 \]Problem 2: Determine if \( g(x) \) is continuous at \( x = 1 \):
\[ g(x) = \begin{cases} 3x + 1, & x < 1 \\ 4, & x = 1 \\ x^2 + 2, & x > 1 \end{cases} \]Solution:
Check left and right-hand limits and compare with \( g(1) \).
The function is not continuous at \( x = 1 \) because \( \lim\limits_{x \to 1^-} g(x) \neq \lim\limits_{x \to 1^+} g(x) \).