Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{x \rightarrow 5^{-}} \frac{x^{2}}{(x-5)(3-x)} $$

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Understand the Expression and Check Initial Substitution

We are finding the limit of the expression \( \frac{x^2}{(x-5)(3-x)} \) as \( x \) approaches 5 from the left (5\(^-\)). First, substitute \( x = 5 \) into the expression to check for any obvious issues or simplification opportunities. Substituting 5 directly gives: \( \frac{25}{(5-5)(3-5)} = \frac{25}{0} \), which is undefined. This means we need another approach to evaluate the limit.

Step 2: Simplify the Expression If Possible

Before finding a limit where some undefined points occur, it might help to analyze and simplify the expression. However, in this expression, there is no obvious simplification since both \((x-5)\) and \((3-x)\) are linear terms related by a factor (-1) with \( x \approx 5 \).

Step 3: Analyze the Expression Behavior Near \( x = 5 \)

We need to consider the behavior of the numerator and denominator as \( x \to 5^- \). The term \((x-5)\) approaches 0 from the negative side since x is slightly less than 5. Meanwhile, the term \((3-x)\) is positive because 3 is less than any number close to 5. The numerator \(x^2\) approaches 25, a constant positive number.

Step 4: Evaluate Limit with Sign Analysis

As \( x \to 5^- \), the expression \( \frac{x^2}{(x-5)(3-x)} \) their behaviors can be checked: the numerator remains positive (going towards 25), \((x-5)\) goes to zero negatively (approaching zero from the left), and \((3-x)\) is positive, making the entire denominator approach zero from the positive. Consequently, the expression becomes a large negative number.

Step 5: Conclusion of the Limit

By analyzing the component terms and their signs as \( x \to 5^- \), we determine: the expression tends to \(-\infty\) as a positive number divided by a negative small number results in a large negative value. Thus, \[\lim_{x \to 5^-} \frac{x^2}{(x-5)(3-x)} = -\infty.\]

Key Concepts

Limit Evaluation

When dealing with limits, our goal is to understand the behavior of a function as it approaches a certain value. The expression given is \( \lim_{x \to 5^-} \frac{x^2}{(x-5)(3-x)} \). The first step in limit evaluation is to substitute the approaching value, which is 5 in this case. However, upon substitution, we find \( \frac{25}{0} \), which is undefined. This tells us that mere substitution won't work, and a deeper analysis is needed. Evaluating limits involves understanding how the function behaves near the point of interest, especially if direct substitution results in undefined forms. The goal is to use this analysis to correctly identify how the function behaves as \( x \) nears the target value.

Behavior Near Singularities

Singularities occur where a function isn't defined or behaves indefinitely, like near \( x = 5 \) in our expression. Understanding this behavior includes examining the components of the function as \( x \to 5^- \). As \( x \) nears 5 from the left, the term \((x-5)\) gets closer to zero but remains negative, while the term \((3-x)\) stays positive since 3 is less than numbers close to 5. These key indicators give us clues about the singular nature of the expression. Recognizing the behavior near singularities helps avoid pitfalls of indeterminate values and allows us to accurately estimate the limit's behavior.

Sign Analysis

Sign analysis is vital when evaluating limits to determine the behavior of a function around singular points. In the expression \( \frac{x^2}{(x-5)(3-x)} \), each component's sign plays a crucial role. As \( x \to 5^- \), \( x^2 \) remains positive; \((x-5)\) is negative as it approaches zero, and \((3-x)\) stays positive. This combination leads to a negative denominator because a positive multiplied by a negative is negative. Hence, the entire expression tends towards a large negative value since a positive numerator over a negative denominator means the result will be negative. Sign analysis helps predict whether the limit will tend towards positive or negative infinity or zero.

Undefined Expressions

An undefined expression often arises when a function behaves unpredictably, such as division by zero which happens in our given expression. This undefined nature is evident from substituting \( x = 5 \) into \( \frac{x^2}{(x-5)(3-x)} \), resulting in zero in the denominator. Undefined expressions require alternative methods like factoring, rationalizing, or assessing terms’ behavior to understand how they affect the limit. Evaluating the limit of an undefined expression demands careful analysis, as ignoring its undefined nature can lead to incorrect conclusions about the function's behavior. Recognizing and resolving undefined expressions is key to mastering calculus limits.