Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 2^{-}} \frac{x^{2}-x-2}{|x-2|} $$

Step-by-Step Solution

Answer

The limit is -3.

Step 1: Recognize the Limit Type

The limit we are dealing with is a one-sided limit from the left (denoted by the "+" in the notation). It is important to consider the direction when evaluating limits involving absolute values as it can affect the expression's behavior.

Step 2: Factor and Analyze the Numerator

The numerator of the expression is a quadratic equation: \(x^2 - x - 2\). We need to factor it to simplify the expression. This can be written as \((x-2)(x+1)\).

Step 3: Simplify the Expression

After factoring the numerator, we substitute it back into the limit expression:\[ \lim_{x \to 2^{-}} \frac{(x-2)(x+1)}{|x-2|} \]Since \(x\) approaches 2 from the left, \(|x-2| = -(x-2)\) because \(x-2\) is negative.

Step 4: Substitute and Simplify Further

Replace \(|x-2|\) with \(-(x-2)\) in the expression, we get:\[ \lim_{x \to 2^{-}} \frac{(x-2)(x+1)}{-(x-2)} \]Cancel out the \((x-2)\) terms in the numerator and denominator:\[ \lim_{x \to 2^{-}} -(x+1) \]

Step 5: Evaluate the Limit

Now, substitute \(x = 2\) directly into the simplified expression:\[ -(2+1) = -3 \]Thus, the limit evaluates to \(-3\).

Key Concepts

One-sided Limits

In calculus, a one-sided limit refers to the behavior of a function as the input approaches a specific value from one side—either from the left or the right. In the expression \( \lim_{x \to 2^{-}} f(x) \), the notation "\(2^{-}\)" indicates that \(x\) approaches 2 from the left. Knowing the direction is crucial, especially for expressions that involve absolute values or non-continuous functions.

One-sided limits are helpful when dealing with piecewise functions or absolute values, where the function's behavior might differ slightly depending on the direction from which \(x\) approaches the point. It allows us to evaluate the limit without confusion about which direction the value is approaching from, providing a clear pathway for calculating accurate results.

Factoring in Calculus

Factoring involves breaking down a complex expression into simpler components, making it easier to handle in calculus problems. In the problem we worked with, the quadratic expression \(x^2 - x - 2\) can be factored into \((x-2)(x+1)\). This simplifies the limit expression and allows us to cancel out terms efficiently, leading to an easier calculation.

Here's how factoring is often used to tackle limits:

• Identify the polynomial that may need to be factored, such as quadratics.
• Apply appropriate factoring techniques, either by inspection or using methods such as the quadratic formula, if necessary.
• Simplify the expression by canceling common factors in the numerator and denominator.

By reducing the complexity of the initial expression, factoring helps in isolating problematic terms, streamlining the path towards limit evaluation.

Absolute Value in Limits

Absolute values add an interesting twist to limits, as they account for the magnitude of a number while disregarding its sign. In the limit expression \(\frac{(x-2)(x+1)}{|x-2|}\), the absolute value \(|x-2|\) ensures a non-negative denominator.

For one-sided limits, handling absolute values requires attention to the direction from which the value is approached. When \(x \to 2^{-}\), \(x-2\) is negative, making \(|x-2| = -(x-2)\). This sign change is critical for simplifying the expression correctly.

• Consider whether the approach to the limit makes the absolute value positive or negative.
• Substitute known negative or positive form of the expression when simplifying.

Understanding absolute value in the context of limits ensures correct algebraic manipulation and proper limit evaluation.

Limit Evaluation Techniques

Several techniques exist for evaluating limits, each suited to certain types of problems. Choosing the right one depends on the nature of the expression at hand. In our problem, key techniques included identifying the type of limit, factoring, simplification, and direct substitution.

Steps for evaluating limits include:

• Analyze the type of limit: Confirm if one-sided or two-sided.
• Factor expressions where applicable to simplify cancellations.
• Handle absolute values appropriately based on the approach direction.
• Once simplified, substitute the approach value to find the limit.

By following these steps, mathematicians often transform complex expressions into straightforward calculations, finding the exact limiting value as approached by the variable. This structure boosts confidence in tackling challenging limit problems effectively.