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A vertical asymptote is a line that a graph approaches but never actually touches or crosses as it extends vertically to infinity or negative infinity. In calculus, finding a vertical asymptote involves looking at the behavior of a function as it nears certain x-values.

With the function given in the exercise, \( f(x) = \left| \frac{\sin(1/x)}{x} \right| \), we identified that there is a vertical asymptote at \( x = 0 \). This was determined by examining how \( f(x) \) behaves as \( x \) approaches 0 from both sides:

• As \( x \to 0^+ \), the function \( f(x) \) approaches infinity.
• As \( x \to 0^- \), the function \( f(x) \) also approaches infinity.

Since \( f(x) \) increases without bound as it nears \( x = 0 \) from either direction, this corroborates the presence of a vertical asymptote at \( x = 0 \).

This concept helps students understand how certain operations, like these involving division by zero, play a critical role in defining the overall behavior of functions in calculus.

The domain of a function refers to all the possible input values (x-values) that allow the function to operate without resolving into an undefined or problematic state. Establishing the domain is essential because it tells us over which intervals a function is properly defined.

For \( f(x) = \left| \frac{\sin(1/x)}{x} \right| \), both the sine function \( \sin(1/x) \) and the division by \( x \) are cause for careful domain analysis:

• Division by \( x \) means that the function is undefined at \( x = 0 \), because dividing by zero is undefined.
• Hence, despite \( \sin(1/x) \) being defined for non-zero \( x \), division makes the entire function undefined at \( x = 0 \).

Therefore, the domain of \( f(x) \) is all real numbers except \( x = 0 \), or in interval notation, \( (-\infty, 0) \cup (0, \infty) \).

Understanding the domain helps in sketching graphs and predicting behaviors in contexts where certain x-values are involved.

Oscillation refers to the repeated and rapid fluctuation of a function's values between definite bounds, in this case between -1 and 1 for the sine function. Identifying oscillations in a function is crucial as it informs us about its stability and behavior near particular values.

In the given function \( f(x) = \left| \frac{\sin(1/x)}{x} \right| \), the term \( \sin(1/x) \) is responsible for oscillation:

• As \( x \) approaches 0, the frequency of \( \sin(1/x) \) increases, meaning it oscillates more rapidly between -1 and 1.
• This rapid fluctuation results in \( \frac{\sin(1/x)}{x} \) fluctuating between extreme positive and negative values.
• Since the oscillations do not diminish in magnitude as \( x \to 0 \), the amplitude grows unbounded.

The combination of oscillation and an undefined factor at \( x = 0 \) contributes to the formation of a vertical asymptote there. Recognizing patterns of oscillation, especially in trigonometric functions, is essential in calculus for graphing and analyzing function behaviors.