Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x sin (1/x) = 0

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Understand the formal definition of a limit: For the limit \( \lim_{x \to a} f(x) = L \), for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).

Identify the function and limit in the problem: Here, \( f(x) = x \sin(1/x) \) and we want to prove \( \lim_{x \to 0} x \sin(1/x) = 0 \).

Express \( |f(x) - L| \) using the given function and limit: Since \( L = 0 \), we have \( |x \sin(1/x) - 0| = |x \sin(1/x)| \).

Use the property of the sine function: Since \( |\sin(1/x)| \leq 1 \) for all \( x \neq 0 \), it follows that \( |x \sin(1/x)| \leq |x| \).

Choose \( \delta = \epsilon \) to satisfy the limit definition: For \( |x| < \delta \), we have \( |x \sin(1/x)| \leq |x| < \epsilon \), thus proving the limit statement.

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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.

Squeeze Theorem

The Squeeze Theorem is a method used to find limits of functions that are difficult to evaluate directly. It states that if f(x) ≤ g(x) ≤ h(x) for all x in some interval around a (except possibly at a), and if the limits of f(x) and h(x) as x approaches a are both L, then the limit of g(x) as x approaches a is also L. This theorem is particularly useful for functions like x sin(1/x).

Behavior of Oscillating Functions

The function sin(1/x) oscillates between -1 and 1 as x approaches 0, which means it does not settle at a single value. However, when multiplied by x, which approaches 0, the product x sin(1/x) is squeezed to 0. Understanding the behavior of oscillating functions is essential for analyzing limits involving such terms.

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