Analyzing Limits And Asymptotes Of G(x) = (x^2 - 16) / (3x^2 + 27x + 60)

In this article, we will delve into the intricacies of the function g(x) = (x^2 - 16) / (3x^2 + 27x + 60). Our primary focus will be on evaluating the limit of g(x) as x approaches -4 and determining the presence of any vertical asymptotes at x = -4. This involves a detailed analysis of the function's behavior around the specified point and a thorough understanding of the concepts of limits and asymptotes in calculus. We will break down the function, factorize the numerator and denominator, and identify any common factors that might lead to removable discontinuities. Furthermore, we will explore the implications of these findings on the overall characteristics of the function, providing a comprehensive understanding of its behavior and properties.

Understanding the Function g(x)

To begin our analysis, let's first express the function g(x) in its most simplified form. The given function is g(x) = (x^2 - 16) / (3x^2 + 27x + 60). The initial step involves factoring both the numerator and the denominator to identify any common factors. This process is crucial for understanding the function's behavior and identifying any potential discontinuities. Factoring allows us to simplify the expression and reveal any hidden properties that might not be immediately apparent. This is a fundamental technique in calculus and is essential for analyzing rational functions. By carefully factoring both the numerator and denominator, we can gain valuable insights into the function's nature, its limits, and the presence of any asymptotes.

The numerator, x^2 - 16, is a difference of squares and can be factored as (x - 4)(x + 4). This factorization is a standard algebraic identity and is a crucial step in simplifying the function. Recognizing the difference of squares pattern allows for a quick and efficient factorization, which is essential for further analysis. The factors (x - 4) and (x + 4) will play a significant role in determining the function's behavior, especially around the points where these factors equal zero. Understanding these factors is key to identifying any potential vertical asymptotes or removable discontinuities.

The denominator, 3x^2 + 27x + 60, can be simplified by first factoring out the common factor of 3, resulting in 3(x^2 + 9x + 20). This simplification makes the subsequent factorization easier. Factoring out the common factor is a crucial step in simplifying the expression and revealing the underlying quadratic. The simplified quadratic expression, x^2 + 9x + 20, can then be factored into (x + 4)(x + 5). This factorization is a standard quadratic factorization technique and is essential for further analysis. Combining the common factor of 3 with the factored quadratic gives us the complete factorization of the denominator as 3(x + 4)(x + 5). This factorization is crucial for identifying the function's vertical asymptotes and removable discontinuities.

Therefore, the factored form of the function g(x) is g(x) = ((x - 4)(x + 4)) / (3(x + 4)(x + 5)). This factored form is crucial for our subsequent analysis, as it allows us to identify common factors and simplify the expression further. The factored form reveals the presence of the common factor (x + 4) in both the numerator and the denominator, which indicates a potential removable discontinuity at x = -4. This observation is critical for evaluating the limit of the function as x approaches -4. The factored form also helps in identifying the vertical asymptotes of the function, which occur at the values of x that make the denominator zero but do not cancel out with factors in the numerator. By carefully examining the factored form, we can gain a deeper understanding of the function's behavior and its key characteristics.

Evaluating the Limit as x Approaches -4

Now, let's evaluate the limit of g(x) as x approaches -4. From the factored form g(x) = ((x - 4)(x + 4)) / (3(x + 4)(x + 5)), we observe a common factor of (x + 4) in both the numerator and denominator. This common factor suggests the presence of a removable discontinuity at x = -4. To evaluate the limit, we can cancel out this common factor, provided that x ≠ -4. This simplification is a crucial step in determining the limit, as it removes the indeterminate form that arises when directly substituting x = -4 into the original function. Canceling out the common factor allows us to work with a simplified expression that is equivalent to the original function everywhere except at x = -4.

After canceling the common factor, the simplified function becomes (x - 4) / (3(x + 5)). This simplified form is equivalent to the original function for all values of x except x = -4. Now, we can directly substitute x = -4 into this simplified expression to evaluate the limit. Direct substitution is a fundamental technique in evaluating limits, and it is applicable when the function is continuous at the point where the limit is being taken. In this case, the simplified function is continuous at x = -4, allowing us to directly substitute the value and obtain the limit.

Substituting x = -4 into the simplified expression, we get (-4 - 4) / (3(-4 + 5)) = -8 / 3. This calculation gives us the value of the limit as x approaches -4. The result, -8/3, is a finite number, which indicates that the function has a removable discontinuity at x = -4 and the limit exists at this point. This finding is crucial for understanding the function's behavior around x = -4. The limit represents the value that the function approaches as x gets arbitrarily close to -4, even though the function itself is not defined at that exact point.

Therefore, lim x→-4 g(x) = -8/3, which is approximately -2.667. This result confirms that the function approaches a specific value as x gets closer to -4. The numerical value of the limit provides a quantitative measure of the function's behavior around the point of discontinuity. The fact that the limit exists and is finite indicates that the discontinuity is removable, meaning that the function can be redefined at x = -4 to make it continuous. This understanding is essential for various applications of calculus, such as curve sketching and optimization problems.

Determining Vertical Asymptotes

Next, let's determine if g(x) has a vertical asymptote at x = -4. A vertical asymptote occurs at a value of x where the function approaches infinity or negative infinity. In the factored form g(x) = ((x - 4)(x + 4)) / (3(x + 4)(x + 5)), we identified a common factor of (x + 4), which we canceled out when evaluating the limit. This cancellation indicates that there is a removable discontinuity at x = -4, not a vertical asymptote. A vertical asymptote occurs when a factor in the denominator remains after simplification and makes the denominator equal to zero. In this case, the factor (x + 4) was canceled out, so it does not contribute to a vertical asymptote at x = -4.

After canceling the common factor, the simplified denominator is 3(x + 5). This simplified denominator helps us identify the vertical asymptotes of the function. A vertical asymptote occurs at the values of x that make the simplified denominator equal to zero but do not make the numerator zero. Setting the simplified denominator equal to zero, we get 3(x + 5) = 0, which implies x = -5. This value of x is a potential vertical asymptote because it makes the denominator zero while the numerator is non-zero.

At x = -5, the simplified numerator is (-5 - 4) = -9, which is non-zero. This confirms that there is a vertical asymptote at x = -5. The presence of a vertical asymptote at x = -5 indicates that the function approaches infinity or negative infinity as x gets closer to -5. This behavior is a characteristic feature of vertical asymptotes and is important for understanding the function's overall shape and behavior. The vertical asymptote at x = -5 divides the domain of the function into intervals where the function's behavior can be analyzed separately.

Therefore, g(x) does not have a vertical asymptote at x = -4 because the factor (x + 4) was canceled out. Instead, there is a removable discontinuity at this point. The function has a vertical asymptote at x = -5, which is where the simplified denominator becomes zero. This analysis highlights the importance of factoring and simplifying the function to accurately identify its key features, such as removable discontinuities and vertical asymptotes. Understanding these features is crucial for a comprehensive understanding of the function's behavior and properties.

Conclusion

In conclusion, the limit of g(x) as x approaches -4 is -8/3 (approximately -2.667), and g(x) does not have a vertical asymptote at x = -4. The function has a removable discontinuity at x = -4 due to the common factor (x + 4) in the numerator and denominator. The vertical asymptote of the function is located at x = -5. This comprehensive analysis demonstrates the importance of factoring, simplifying, and evaluating limits to understand the behavior of rational functions. By carefully examining the factored form of the function, we can identify key features such as removable discontinuities and vertical asymptotes, which are essential for a complete understanding of the function's properties and behavior.