Problem 40 - Calculus: Early Transcendentals

Question

Determine the infinite limit. \( \displaystyle \lim_{x \to 2^-}\frac{x^2 - 2x}{x^2 - 4x + 4} \)

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Identify the Limit Type

We're asked to determine the limit as \(x\) approaches \(2\) from the left of the function \(\frac{x^2 - 2x}{x^2 - 4x + 4}\). This is a left-hand limit, denoted by \(x \to 2^-\). We need to examine the behavior of the function near \(x = 2\).

Step 2: Simplify the Expression

Observe that the denominator \(x^2 - 4x + 4\) can be factored. Notice it is a perfect square: \((x - 2)^2\). The numerator \(x^2 - 2x\) can be factored as \(x(x - 2)\). Thus, the expression simplifies to \(\frac{x(x - 2)}{(x - 2)^2}\).

Step 3: Cancel Common Factors

In \(\frac{x(x - 2)}{(x - 2)^2}\), the \((x - 2)\) term is common in both the numerator and the denominator. We cancel \((x - 2)\) from both, simplifying the expression to \(\frac{x}{x - 2}\), noting that \(x = 2\) would still be undefined.

Step 4: Analyze the Behavior as x Approaches 2 from the Left

As \(x\) approaches \(2\) from the left in the expression \(\frac{x}{x-2}\), \(x - 2\) becomes a small negative number, causing the entire fraction to become very negative (since numerator is 2, a positive number). Thus, the function approaches negative infinity.

Step 5: Conclusion

Therefore, the limit of the function as \(x\) approaches 2 from the left is \(-\infty\).

Key Concepts

Left-hand Limits

When evaluating left-hand limits, we focus on what happens to a function as the variable approaches a certain value from the left side of the number line. Specifically, for the limit \( \lim_{x \to 2^-} f(x) \), our interest lies in how \( f(x) \) behaves as \( x \) inches closer to 2 from values less than 2. This requires us to closely analyze the function’s trend\( \) just before reaching the value in question, which may not be the function’s behavior at exactly that point.

• The notation \( x \to 2^- \) signifies approaching from values less than 2.
• It's essential to examine how the function behaves, not just computationally compute values.
• In this problem, as \( x \to 2^- \), the behavior of the fraction is key to understanding its limit.

Exploring left-hand limits is fundamental in calculus for cases where the function exhibits different behaviors on either side of a given point.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of simpler polynomials. This step can simplify complex expressions greatly, especially when dealing with functions that involve limits or continuity.

• This technique is used to break down the given polynomial into smaller, manageable parts.
• For our expression \( x^2 - 4x + 4 \), we recognized it as \((x-2)^2\). This reveals critical points where variables might be canceled to simplify the limit expression.
• The numerator, \( x^2 - 2x \), similarly factors to \( x(x - 2)\).

Breaking down polynomials into their factors helps identify common terms that might simplify the overall expression involved in limits or other calculus operations.

Simplifying Expressions

Simplifying expressions is imperative when solving limit problems, as it helps to see a clearer picture of how functions behave. After factoring the expression \( \frac{x^2 - 2x}{x^2 - 4x + 4} \) to \( \frac{x(x - 2)}{(x-2)^2} \), the next step is simplification. By canceling the common term \( (x-2) \), we obtain \( \frac{x}{x-2} \).

• Always look for common factors or patterns; this often reduces complex expressions.
• Ensure that when canceling terms, the domain constraints remain intact (i.e., check where any terms originally posed division by zero).

This simplification ensures our expression is in a form that easily demonstrates behavior as \( x \) approaches the value of interest. It’s the simplification that reveals the core behavior of the function near critical points.

Analyzing Function Behavior

In calculus, understanding how a function behaves near a particular value is crucial for limit evaluation. Analyzing function behavior means understanding the outcome of the function as it approaches the critical point.

• For our simplified function \( \frac{x}{x-2} \), as \( x \to 2^- \), \( x-2 \) approaches a tiny negative value. This makes the expression near \(-2^+\).
• Therefore, as \( x \) reaches closer to 2 from the left, the function produces increasingly large negative values, resulting in \(-\infty\).
• This behavior highlights the importance of analyzing both the numerator and the denominator as \( x \) approaches the value, ensuring that even canceled terms don't skew our understanding.

Analyzing function behavior is crucial because it helps make an informed conclusion about the limit of the function, shedding light on its behavior near discontinuities or undefined points.