Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 0 (1 / |x|) = ∞

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Step 1: Understand the formal definition of a limit approaching infinity. The statement lim x → 0 (1 / |x|) = ∞ means that for every positive number M, there exists a δ > 0 such that if 0 < |x| < δ, then 1 / |x| > M.

Step 2: Start by choosing an arbitrary positive number M. Our goal is to find a δ > 0 such that whenever 0 < |x| < δ, the inequality 1 / |x| > M holds true.

Step 3: Rearrange the inequality 1 / |x| > M to find a condition on |x|. This gives |x| < 1 / M. This suggests that we can choose δ = 1 / M.

Step 4: Verify the choice of δ. If 0 < |x| < δ, then 0 < |x| < 1 / M, which implies 1 / |x| > M. This satisfies the condition required by the definition of the limit.

Step 5: Conclude the proof by stating that for every positive number M, we have found a δ = 1 / M such that if 0 < |x| < δ, then 1 / |x| > M, thus proving lim x → 0 (1 / |x|) = ∞ using the formal definition.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit Definition

The formal definition of a limit states that for a function f(x), the limit as x approaches a value 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.

Behavior of Functions Near Zero

Understanding how functions behave as they approach a specific point, particularly zero, is essential. For the function f(x) = 1/|x|, as x approaches 0 from either side, the function values increase without bound, indicating that the limit approaches infinity.

Infinity in Limits

In calculus, stating that a limit equals infinity means that the function grows larger and larger without bound as it approaches a certain point. This concept is vital for interpreting limits like lim x → 0 (1 / |x|) = ∞, as it signifies that the function does not settle at a finite value but rather diverges.

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