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Question
Which functions are continuous at x equals negative 2? Select all that apply.
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f left-parenthesis x right-parenthesis equals enlarged left-brace Start Layout 1st row Start Fraction x squared minus 4 over x plus 2 End Fraction if x not equal to negative 2 2nd row negative 4 if x equals negative 2 End Layout
Image with alt text: f left-parenthesis x right-parenthesis equals enlarged left-brace Start Layout 1st row Start Fraction x squared minus 4 over x plus 2 End Fraction if x not equal to negative 2 2nd row negative 4 if x equals negative 2 End Layout

f left-parenthesis x right-parenthesis equals enlarged left-brace Start Layout 1st row Start Fraction x squared minus 4 over x plus 2 End Fraction if x not equal to negative 2 2nd row 0 if x equals negative 2 End Layout
Image with alt text: f left-parenthesis x right-parenthesis equals enlarged left-brace Start Layout 1st row Start Fraction x squared minus 4 over x plus 2 End Fraction if x not equal to negative 2 2nd row 0 if x equals negative 2 End Layout

f left-parenthesis x right-parenthesis equals enlarged left-brace Start Layout 1st row Start Fraction x squared plus 4 x plus 4 over x plus 2 End Fraction if x not equal to negative 2 2nd row 0 if x equals negative 2 End Layout
Image with alt text: f left-parenthesis x right-parenthesis equals enlarged left-brace Start Layout 1st row Start Fraction x squared plus 4 x plus 4 over x plus 2 End Fraction if x not equal to negative 2 2nd row 0 if x equals negative 2 End Layout

f left-parenthesis x right-parenthesis equals enlarged left-brace Start Layout 1st row Start Fraction x squared plus 4 x plus 4 over x plus 2 End Fraction if x not equal to negative 2 2nd row 4 if x equals negative 2 End Layout
Image with alt text: f left-parenthesis x right-parenthesis equals enlarged left-brace Start Layout 1st row Start Fraction x squared plus 4 x plus 4 over x plus 2 End Fraction if x not equal to negative 2 2nd row 4 if x equals negative 2 End Layout
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1 answer

To determine which functions are continuous at \( x = -2 \), we need to check the continuity condition:

A function \( f(x) \) is continuous at \( x = a \) if:

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

We will analyze each given function:

  1. Function:
    \[ f(x) = \begin{cases} \frac{x^2 - 4}{x + 2} & \text{if } x \neq -2 \ -4 & \text{if } x = -2 \end{cases} \]

    • For \( x \neq -2 \), \( f(x) = \frac{x^2 - 4}{x + 2} = \frac{(x-2)(x+2)}{x+2} = x - 2 \) (for \( x \neq -2 \)).
    • The limit as \( x \to -2 \) is \( -2 - 2 = -4 \).
    • Since \( \lim_{x \to -2} f(x) = -4 \) and \( f(-2) = -4 \), this function is continuous at \( x = -2 \).

  2. Function:
    \[ f(x) = \begin{cases} \frac{x^2 - 4}{x + 2} & \text{if } x \neq -2 \ 0 & \text{if } x = -2 \end{cases} \]

    • The limit as \( x \to -2 \) is \( -4 \).
    • Since \( \lim_{x \to -2} f(x) = -4 \) and \( f(-2) = 0 \), this function is not continuous at \( x = -2 \).

  3. Function:
    \[ f(x) = \begin{cases} \frac{x^2 + 4x + 4}{x + 2} & \text{if } x \neq -2 \ 0 & \text{if } x = -2 \end{cases} \]

    • For \( x \neq -2 \), \( f(x) = \frac{(x + 2)^2}{x + 2} = x + 2 \) (for \( x \neq -2 \)).
    • The limit as \( x \to -2 \) is \( -2 + 2 = 0 \).
    • Since \( \lim_{x \to -2} f(x) = 0 \) and \( f(-2) = 0 \), this function is continuous at \( x = -2 \).

  4. Function:
    \[ f(x) = \begin{cases} \frac{x^2 + 4x + 4}{x + 2} & \text{if } x \neq -2 \ 4 & \text{if } x = -2 \end{cases} \]

    • The limit as \( x \to -2 \) is \( 0 \).
    • Since \( \lim_{x \to -2} f(x) = 0 \) and \( f(-2) = 4 \), this function is not continuous at \( x = -2 \).

Final Answer: The functions that are continuous at \( x = -2 \) are:

  • The first function
  • The third function