Use the formal definitions from Exercise 97 to prove the limit statements in Exercises 98–102.
lim x→2⁻ (1 / (x − 2)) = −∞

검증된 단계별 안내

Understand the problem: We need to prove that as x approaches 2 from the left (x → 2⁻), the function f(x) = 1 / (x - 2) approaches negative infinity.

Recall the formal definition of a limit approaching negative infinity: For every positive number M, there exists a δ > 0 such that if 0 < |x - 2| < δ and x < 2, then f(x) < -M.

Set up the inequality: We want to show that 1 / (x - 2) < -M. This implies that x - 2 < -1/M, which further implies x < 2 - 1/M.

Choose δ: To satisfy the condition 0 < |x - 2| < δ and x < 2, we can choose δ = min(1, 1/M). This ensures that x is close to 2 from the left and satisfies the inequality.

Conclude the proof: With this choice of δ, whenever 0 < |x - 2| < δ and x < 2, the inequality 1 / (x - 2) < -M holds, thus proving that the limit of f(x) as x approaches 2 from the left is negative infinity.

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Limit Definition

The limit of a function at a point describes the behavior of the function as it approaches that point. Formally, we say that the limit of f(x) as x approaches a is L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε. This definition is crucial for proving limit statements rigorously.

One-Sided Limits

One-sided limits consider the behavior of a function as it approaches a specific point from one side only. The left-hand limit, denoted as lim x→a⁻ f(x), examines the values of f(x) as x approaches a from the left. Understanding one-sided limits is essential for analyzing functions that exhibit different behaviors from either side of a point, particularly in cases of discontinuity.

Behavior of Rational Functions

Rational functions are ratios of polynomials, and their limits can often lead to infinite values. In the case of lim x→2⁻ (1 / (x - 2)), as x approaches 2 from the left, the denominator approaches zero negatively, causing the function to decrease without bound. Recognizing how rational functions behave near their vertical asymptotes is key to understanding limits that result in ±∞.

Use the formal definitions from Exercise 97 to prove the limit statements in Exercises 98–102.lim x→2⁻ (1 / (x − 2)) = −∞