Calculating Limits Lim X To -6+ And Lim X To -6- (2x + 12) / |x + 6|

In the realm of calculus, limits play a fundamental role in understanding the behavior of functions as their input approaches a specific value. This article delves into the calculation of two specific limits: ${\lim_{x \to -6^+} \frac{2x + 12}{|x + 6|}}$ and ${\lim_{x \to -6^-} \frac{2x + 12}{|x + 6|}}$. These limits explore the function's behavior as ${x}$ approaches -6 from the right (positive side) and from the left (negative side), respectively. Understanding these one-sided limits provides valuable insights into the function's overall behavior near the point ${x = -6}$.

Understanding the Limit Concept

Before we dive into the calculations, let's briefly revisit the concept of a limit. The limit of a function ${f(x)}$ as ${x}$ approaches a value ${c}$, denoted as ${\lim_{x \to c} f(x)}$, represents the value that ${f(x)}$ gets arbitrarily close to as ${x}$ gets arbitrarily close to ${c}$, but not necessarily equal to ${c}$. In simpler terms, it's the value the function "tends towards" as its input gets closer and closer to a certain point.

One-sided limits, as the name suggests, consider the function's behavior as ${x}$ approaches ${c}$ from either the right (denoted by ${x \to c^+}$) or the left (denoted by ${x \to c^-}$). These are crucial when dealing with functions that may behave differently depending on the direction of approach.

Analyzing ${\lim_{x \to -6^+} \frac{2x + 12}{|x + 6|}}$

Let's begin by evaluating the limit as ${x}$ approaches -6 from the right: ${\lim_{x \to -6^+} \frac{2x + 12}{|x + 6|}}$. This means we are considering values of ${x}$ that are slightly greater than -6.

When analyzing limits, the first step is often to attempt direct substitution. However, in this case, substituting ${x = -6}$ directly into the expression results in an indeterminate form of ${\frac{0}{0}}$, which doesn't give us the limit's value. This indicates that we need to manipulate the expression algebraically to simplify it and remove the indeterminacy.

Our key insight here lies in the absolute value function, ${|x + 6|}$. When ${x}$ approaches -6 from the right (i.e., ${x > -6}$), the expression ${x + 6}$ is positive. Therefore, the absolute value ${|x + 6|}$ is simply equal to ${x + 6}$ in this case. We can rewrite the limit as:

${\lim_{x \to -6^+} \frac{2x + 12}{|x + 6|} = \lim_{x \to -6^+} \frac{2x + 12}{x + 6}}$

Now, we can factor out a 2 from the numerator:

${\lim_{x \to -6^+} \frac{2(x + 6)}{x + 6}}$

Since we are considering the limit as ${x}$ approaches -6, but not actually equal to -6, we can cancel the ${(x + 6)}$ terms in the numerator and denominator:

${\lim_{x \to -6^+} \frac{2(x + 6)}{x + 6} = \lim_{x \to -6^+} 2}$

This leaves us with a constant limit, which is simply the constant value itself:

${\lim_{x \to -6^+} 2 = 2}$

Therefore, the limit of the function as ${x}$ approaches -6 from the right is 2. This means that as ${x}$ gets closer and closer to -6 from values greater than -6, the function ${\frac{2x + 12}{|x + 6|}}$ approaches the value 2.

Analyzing ${\lim_{x \to -6^-} \frac{2x + 12}{|x + 6|}}$

Next, we will evaluate the limit as ${x}$ approaches -6 from the left: ${\lim_{x \to -6^-} \frac{2x + 12}{|x + 6|}}$. This means we are now considering values of ${x}$ that are slightly less than -6.

Again, the key to solving this limit lies in understanding the behavior of the absolute value function. When ${x}$ approaches -6 from the left (i.e., ${x < -6}$), the expression ${x + 6}$ is negative. Therefore, the absolute value ${|x + 6|}$ is equal to the negation of ${x + 6}$, which is ${-(x + 6)}$. This is because the absolute value function always returns a non-negative value.

We can rewrite the limit as:

${\lim_{x \to -6^-} \frac{2x + 12}{|x + 6|} = \lim_{x \to -6^-} \frac{2x + 12}{-(x + 6)}}$

Factoring out a 2 from the numerator, we get:

${\lim_{x \to -6^-} \frac{2(x + 6)}{-(x + 6)}}$

As before, since we are considering the limit as ${x}$ approaches -6, but not actually equal to -6, we can cancel the ${(x + 6)}$ terms:

${\lim_{x \to -6^-} \frac{2(x + 6)}{-(x + 6)} = \lim_{x \to -6^-} -2}$

This again leaves us with a constant limit:

${\lim_{x \to -6^-} -2 = -2}$

Therefore, the limit of the function as ${x}$ approaches -6 from the left is -2. This signifies that as ${x}$ gets closer and closer to -6 from values less than -6, the function ${\frac{2x + 12}{|x + 6|}}$ approaches the value -2.

The Significance of One-Sided Limits

We have now calculated both one-sided limits of the function ${\frac{2x + 12}{|x + 6|}}$ as ${x}$ approaches -6. We found that:

• ${\lim_{x \to -6^+} \frac{2x + 12}{|x + 6|} = 2}$
• ${\lim_{x \to -6^-} \frac{2x + 12}{|x + 6|} = -2}$

The fact that the two one-sided limits are different is crucial. A fundamental theorem in calculus states that the two-sided limit, ${\lim_{x \to c} f(x)}$, exists if and only if both one-sided limits, ${\lim_{x \to c^+} f(x)}$ and ${\lim_{x \to c^-} f(x)}$, exist and are equal. In our case, the one-sided limits exist (2 and -2), but they are not equal. Therefore, we can conclude that the two-sided limit:

${\lim_{x \to -6} \frac{2x + 12}{|x + 6|}}$

does not exist. This indicates a discontinuity in the function's behavior at ${x = -6}$. The function approaches different values depending on the direction from which ${x}$ approaches -6.

Visualizing the Function's Behavior

To further understand this behavior, it can be helpful to visualize the function's graph. The graph of ${y = \frac{2x + 12}{|x + 6|}}$ will show a jump discontinuity at ${x = -6}$. As ${x}$ approaches -6 from the right, the graph will approach the horizontal line ${y = 2}$, and as ${x}$ approaches -6 from the left, the graph will approach the horizontal line ${y = -2}$. There will be a break or jump in the graph at ${x = -6}$, visually demonstrating the non-existence of the two-sided limit.

Conclusion

In this article, we successfully calculated the one-sided limits ${\lim_{x \to -6^+} \frac{2x + 12}{|x + 6|}}$ and ${\lim_{x \to -6^-} \frac{2x + 12}{|x + 6|}}$. By carefully analyzing the behavior of the absolute value function and applying algebraic simplification, we determined that these limits are 2 and -2, respectively. The inequality of these one-sided limits led us to the conclusion that the two-sided limit ${\lim_{x \to -6} \frac{2x + 12}{|x + 6|}}$ does not exist, highlighting a discontinuity in the function's behavior at ${x = -6}$. This exploration underscores the importance of one-sided limits in fully understanding a function's behavior near a point and determining the existence of two-sided limits. This process provides a solid foundation for more advanced calculus concepts, such as continuity and differentiability. Calculating limits is not just a mathematical exercise; it is a powerful tool for analyzing the behavior of functions and understanding the fundamental concepts of calculus. Understanding these nuances is crucial for anyone delving deeper into the world of mathematical analysis.