Limit Of G(x) = (x^2 - 4x + 8) / (3x - 15) As X Approaches 5

Introduction: Unveiling the Behavior of g(x) as x Approaches 5

In the realm of calculus, understanding the behavior of functions as their input approaches a specific value is a fundamental concept. This exploration delves into the function g(x) = (x^2 - 4x + 8) / (3x - 15), focusing on determining its limit as x approaches 5. This seemingly straightforward task unveils the intricacies of limits, particularly when direct substitution leads to an indeterminate form. By meticulously analyzing the function and employing various limit evaluation techniques, we aim to gain a comprehensive understanding of its behavior near x = 5.

Limits play a crucial role in calculus, serving as the foundation for concepts like continuity, derivatives, and integrals. Determining the limit of a function as x approaches a specific value involves investigating the function's output as its input gets arbitrarily close to that value. The limit may or may not exist, and its value can provide valuable insights into the function's behavior. In this case, we are interested in the limit of g(x) as x approaches 5, which we denote as lim (x→5) g(x). To unravel this, we'll first examine the outcome of direct substitution and then consider alternative methods if needed.

The Pitfalls of Direct Substitution: An Encounter with Indeterminacy

Our initial instinct when faced with a limit problem often leads us to direct substitution. This method involves simply plugging the value that x approaches (in this case, 5) directly into the function. Applying direct substitution to our function, g(x) = (x^2 - 4x + 8) / (3x - 15), we get: g(5) = (5^2 - 45 + 8) / (35 - 15) = (25 - 20 + 8) / (15 - 15) = 13 / 0. The result, 13/0, is undefined, signifying that direct substitution fails in this instance. We encounter a division by zero, which is mathematically impermissible. This outcome indicates that the function g(x) has a potential discontinuity at x = 5. It does not, however, definitively mean that the limit does not exist. It merely suggests that we need to employ a different approach to evaluate the limit.

The result of direct substitution highlights a critical aspect of limits: the limit of a function as x approaches a value is not necessarily the same as the function's value at that point. In other words, even if g(5) is undefined, lim (x→5) g(x) may still exist. Direct substitution serves as a preliminary step, and when it yields an indeterminate form like 0/0 or a nonzero number divided by zero, it signals the need for further investigation. The indeterminate form 0/0, in particular, often suggests the possibility of simplifying the function or using techniques like L'Hôpital's Rule to determine the limit. In our case, we obtained a nonzero number divided by zero, which implies a vertical asymptote at x = 5. This vertical asymptote suggests that the function will approach either positive or negative infinity as x gets closer to 5.

Unveiling the Limit: Strategies Beyond Direct Substitution

Given that direct substitution led to an undefined result, we must explore alternative methods to determine lim (x→5) g(x). Several techniques can be employed, depending on the nature of the function and the indeterminate form encountered. These include:

1. Factoring and Simplification: If the function involves rational expressions, factoring the numerator and denominator may reveal common factors that can be canceled out. This simplification can eliminate the discontinuity and allow for direct substitution or other limit evaluation techniques.
2. Rationalizing the Numerator or Denominator: When dealing with expressions involving radicals, rationalizing the numerator or denominator can help eliminate indeterminate forms. This process involves multiplying the expression by a conjugate to remove the radical.
3. L'Hôpital's Rule: This powerful rule applies when the limit results in the indeterminate form 0/0 or ∞/∞. It states that if the limit of f(x) / g(x) as x approaches c is of the form 0/0 or ∞/∞, then the limit is equal to the limit of f'(x) / g'(x) as x approaches c, provided the latter limit exists. L'Hôpital's Rule can be applied repeatedly until the limit can be evaluated.
4. One-Sided Limits: In some cases, the limit may not exist in the traditional sense, but the one-sided limits (the limits as x approaches the value from the left and from the right) may exist. If the one-sided limits are different, the overall limit does not exist. However, examining one-sided limits can still provide valuable information about the function's behavior near the point of interest.

In our specific case, g(x) = (x^2 - 4x + 8) / (3x - 15), we can start by examining factoring and simplification. However, the numerator, x^2 - 4x + 8, does not factor easily. Therefore, we will shift our focus to analyzing the behavior of the function as x approaches 5 from both the left and the right, which will help us understand the one-sided limits.

Exploring One-Sided Limits: Approaching 5 from Both Directions

Since direct substitution resulted in division by zero, it's likely that the limit does not exist in the traditional sense. However, to fully understand the function's behavior near x = 5, we can investigate the one-sided limits. This involves examining what happens to g(x) as x approaches 5 from the left (values less than 5) and as x approaches 5 from the right (values greater than 5). These are denoted as lim (x→5-) g(x) and lim (x→5+) g(x), respectively. The denominator, 3x - 15, becomes 0 when x = 5. This indicates a potential vertical asymptote at x = 5.

As x approaches 5 from the left (x→5-), 3x - 15 will be a small negative number. The numerator, x^2 - 4x + 8, approaches 13 as x approaches 5. Therefore, when x is slightly less than 5, g(x) will be approximately 13 divided by a small negative number, which results in a large negative number. Hence, lim (x→5-) g(x) = -∞. Conversely, as x approaches 5 from the right (x→5+), 3x - 15 will be a small positive number. The numerator still approaches 13. Thus, when x is slightly greater than 5, g(x) will be approximately 13 divided by a small positive number, resulting in a large positive number. Therefore, lim (x→5+) g(x) = +∞. These one-sided limits confirm the presence of a vertical asymptote at x = 5.

Since the one-sided limits are not equal (one is negative infinity, and the other is positive infinity), the overall limit, lim (x→5) g(x), does not exist. The function approaches negative infinity from the left and positive infinity from the right. This behavior is characteristic of a vertical asymptote, where the function shoots off to positive or negative infinity as x approaches a specific value. In the context of this problem, the fact that the one-sided limits diverge to different infinities indicates a significant discontinuity at x = 5.

Conclusion: The Non-Existence of the Limit and the Significance of Vertical Asymptotes

In conclusion, our analysis of the function g(x) = (x^2 - 4x + 8) / (3x - 15) as x approaches 5 reveals that the limit does not exist. Direct substitution resulted in an indeterminate form, signaling the need for further investigation. By examining the one-sided limits, we found that lim (x→5-) g(x) = -∞ and lim (x→5+) g(x) = +∞. The divergence of these one-sided limits to different infinities confirms that the overall limit, lim (x→5) g(x), does not exist. This behavior is a direct consequence of the vertical asymptote present at x = 5.

The existence of a vertical asymptote at x = 5 implies that the function g(x) is unbounded near this point. As x gets closer to 5 from either side, the function's value becomes arbitrarily large (either positively or negatively). This understanding is crucial in various applications of calculus, such as curve sketching, optimization problems, and the analysis of physical systems. The absence of a limit at a point signifies a discontinuity, which can have significant implications for the function's overall behavior and its applicability in modeling real-world phenomena.

This exploration underscores the importance of employing a multifaceted approach when evaluating limits. Direct substitution serves as a valuable initial step, but when it leads to indeterminate forms, other techniques, such as analyzing one-sided limits, are necessary to gain a complete understanding of the function's behavior. Recognizing and understanding the presence of vertical asymptotes is crucial in accurately interpreting the behavior of functions and their limits.