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Limits and Continuity

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:

  1. \( \lim\limits_{x \to a} f(x) \) exists
  2. \( f(a) \) is defined
  3. \( \lim\limits_{x \to a} f(x) = f(a) \)

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 \( \lim\limits_{x \to 1^-} g(x) \neq \lim\limits_{x \to 1^+} g(x) \).

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