Determine the following limits.

b. lim x→3^− 2/(x − 3)^3

Guida verificata passo dopo passo

Step 1: Identify the type of limit. This is a one-sided limit as \( x \) approaches 3 from the left (denoted by \( x \to 3^- \)).

Step 2: Recognize that the expression \( \frac{2}{(x - 3)^3} \) involves a power of \( (x - 3) \) in the denominator, which will approach zero as \( x \to 3^- \).

Step 3: Analyze the behavior of \( (x - 3)^3 \) as \( x \to 3^- \). Since \( x \) is approaching 3 from the left, \( x - 3 \) is negative, and thus \( (x - 3)^3 \) will also be negative and approach zero.

Step 4: Consider the sign of the entire expression \( \frac{2}{(x - 3)^3} \). Since the numerator is positive (2) and the denominator is negative and approaching zero, the overall expression will approach negative infinity.

Step 5: Conclude that the limit is \( -\infty \) as \( x \to 3^- \). This indicates that the function \( \frac{2}{(x - 3)^3} \) decreases without bound as \( x \) approaches 3 from the left.

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit as x approaches 3 from the left (denoted as x→3^−).

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One-Sided Limits

One-sided limits refer to the limits of a function as the input approaches a specific value from one side only, either the left or the right. The notation x→3^− indicates that we are considering the limit as x approaches 3 from values less than 3. This is crucial for analyzing functions that may behave differently from each side of a point.

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One-Sided Limits

Behavior of Rational Functions

Rational functions are ratios of polynomials, and their limits can often be determined by analyzing the behavior of the numerator and denominator as the variable approaches a certain value. In this case, as x approaches 3, the denominator (x - 3)^3 approaches zero, which can lead to infinite limits or undefined behavior, depending on the sign of the numerator.

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