Determine the following limits.


b. lim x→3^− 2/(x − 3)^3

Verified step by step guidance

1. Step 1: Identify the type of limit. This is a one-sided limit as \( x \) approaches 3 from the left (denoted by \( x \to 3^- \)).
2. 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 \( x \to 3^- \).
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.
4. 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.
5. 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.

Key Concepts

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^−).

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.

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.