Question

f(x) = x^2 - 4x - 5

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$$\begin{aligned} &\\lim_{x \\to -\\infty} f(x) \\to -\\infty \\\\ &\\lim_{x \\to \\infty} f(x) \\to -\\infty \\end{aligned}$$

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$$\begin{aligned} &\\lim_{x \\to -\\infty} f(x) \\to \\infty \\\\ &\\lim_{x \\to \\infty} f(x) \\to -\\infty \\end{aligned}$$

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$$\begin{aligned} &\\lim_{x \\to -\\infty} f(x) \\to \\infty \\\\ &\\lim_{x \\to \\infty} f(x) \\to \\infty \\end{aligned}$$

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$$\begin{aligned} &\\lim_{x \\to -\\infty} f(x) \\to -\\infty \\\\ &\\lim_{x \\to \\infty} f(x) \\to \\infty \\end{aligned}$$

Explanation:

Analyze the limit as \(x \to -\infty\)

$$ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (x^2 - 4x - 5) = \infty $$

Analyze the limit as \(x \to \infty\)

$$ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} (x^2 - 4x - 5) = \infty $$

Match with the given options

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Answer:

• (A) \(\lim_{x \to -\infty} f(x) \to -\infty\) and \(\lim_{x \to \infty} f(x) \to -\infty\)
• (B) \(\lim_{x \to -\infty} f(x) \to \infty\) and \(\lim_{x \to \infty} f(x) \to -\infty\)
• (C) \(\lim_{x \to -\infty} f(x) \to \infty\) and \(\lim_{x \to \infty} f(x) \to \infty\) (Correct answer)
• (D) \(\lim_{x \to -\infty} f(x) \to -\infty\) and \(\lim_{x \to \infty} f(x) \to \infty\)