Question

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1. $\lim\limits_{x \to -7^-} f(x) = \square$ help (limits)
2. $\lim\limits_{x \to -7^+} f(x) = \square$
3. $\lim\limits_{x \to -7} f(x) = \square$
4. $\lim\limits_{x \to -5} f(x) = \square$
5. $\lim\limits_{x \to -3^-} f(x) = \square$
6. $\lim\limits_{x \to -3^+} f(x) = \square$
7. $\lim\limits_{x \to -3} f(x) = \square$
8. $\lim\limits_{x \to 2^-} f(x) = \square$
9. $\lim\limits_{x \to 2^+} f(x) = \square$
10. $\lim\limits_{x \to 2} f(x) = \square$
11. $\lim\limits_{x \to +\infty} f(x) = \square$
12. $\lim\limits_{x \to -\infty} f(x) = \square$

Explanation:

To solve these limit problems, we need the graph of the function \( f(x) \) (or a description of its behavior at the given points) because the limit of a function as \( x \) approaches a value depends on the function's behavior near that point (from the left, right, or both sides) and as \( x \) approaches \( \pm\infty \). Since the graph is not provided, I'll explain the general approach for each type of limit:

1. Left-Hand Limit (\( \boldsymbol{\lim_{x \to a^-} f(x)} \))

The left-hand limit as \( x \to a^- \) is the value \( f(x) \) approaches as \( x \) gets closer to \( a \) from the left (values of \( x \) less than \( a \)).

2. Right-Hand Limit (\( \boldsymbol{\lim_{x \to a^+} f(x)} \))

The right-hand limit as \( x \to a^+ \) is the value \( f(x) \) approaches as \( x \) gets closer to \( a \) from the right (values of \( x \) greater than \( a \)).

3. Two-Sided Limit (\( \boldsymbol{\lim_{x \to a} f(x)} \))

The two-sided limit exists if and only if the left-hand limit and right-hand limit at \( x = a \) are equal. If \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \), then \( \lim_{x \to a} f(x) = L \).

4. Limits at Infinity (\( \boldsymbol{\lim_{x \to \infty} f(x)} \) or \( \boldsymbol{\lim_{x \to -\infty} f(x)} \))

These limits describe the end behavior of the function (what \( f(x) \) approaches as \( x \) becomes very large positive or very large negative). For example:

• If \( f(x) \) is a polynomial, the end behavior is determined by the leading term.
• For rational functions, divide numerator and denominator by the highest power of \( x \) in the denominator and analyze the result as \( x \to \pm\infty \).

Next Step

To solve these specific limits, please provide the graph of \( f(x) \) (or a detailed description of its behavior at \( x = -7, -5, -3, 2 \) and as \( x \to \pm\infty \)). Once we have the graph/description, we can:

• Identify the \( y \)-value \( f(x) \) approaches from the left/right at each \( x = a \).
• Check if left and right limits are equal (for two-sided limits).
• Analyze end behavior for \( x \to \pm\infty \).

If you provide the graph or a table of values near these points, I can walk you through each limit calculation!

Answer:

To solve these limit problems, we need the graph of the function \( f(x) \) (or a description of its behavior at the given points) because the limit of a function as \( x \) approaches a value depends on the function's behavior near that point (from the left, right, or both sides) and as \( x \) approaches \( \pm\infty \). Since the graph is not provided, I'll explain the general approach for each type of limit:

1. Left-Hand Limit (\( \boldsymbol{\lim_{x \to a^-} f(x)} \))

The left-hand limit as \( x \to a^- \) is the value \( f(x) \) approaches as \( x \) gets closer to \( a \) from the left (values of \( x \) less than \( a \)).

2. Right-Hand Limit (\( \boldsymbol{\lim_{x \to a^+} f(x)} \))

The right-hand limit as \( x \to a^+ \) is the value \( f(x) \) approaches as \( x \) gets closer to \( a \) from the right (values of \( x \) greater than \( a \)).

3. Two-Sided Limit (\( \boldsymbol{\lim_{x \to a} f(x)} \))

The two-sided limit exists if and only if the left-hand limit and right-hand limit at \( x = a \) are equal. If \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \), then \( \lim_{x \to a} f(x) = L \).

4. Limits at Infinity (\( \boldsymbol{\lim_{x \to \infty} f(x)} \) or \( \boldsymbol{\lim_{x \to -\infty} f(x)} \))

These limits describe the end behavior of the function (what \( f(x) \) approaches as \( x \) becomes very large positive or very large negative). For example:

• If \( f(x) \) is a polynomial, the end behavior is determined by the leading term.
• For rational functions, divide numerator and denominator by the highest power of \( x \) in the denominator and analyze the result as \( x \to \pm\infty \).

Next Step

To solve these specific limits, please provide the graph of \( f(x) \) (or a detailed description of its behavior at \( x = -7, -5, -3, 2 \) and as \( x \to \pm\infty \)). Once we have the graph/description, we can:

• Identify the \( y \)-value \( f(x) \) approaches from the left/right at each \( x = a \).
• Check if left and right limits are equal (for two-sided limits).
• Analyze end behavior for \( x \to \pm\infty \).

If you provide the graph or a table of values near these points, I can walk you through each limit calculation!