Problem 1 - Calculus of a Single Variable

Question

Determining Infinite Limits from a Graph In Exercises \(1-4,\) determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-2\) from the left and from the right. $$ f(x)=2\left|\frac{x}{x^{2}-4}\right| $$

Step-by-Step Solution

Answer

Presuming a graph and checking the values while approaching -2 from both left and right, the function \(f(x)\) approaches \(\infty\) from both directions.

Step 1: Understanding the absolute value

The absolute value of a number always returns a positive value or zero. This will play a crucial role when checking the limit at -2.

Step 2: Checking the limit as x approaches -2 from the left

Starting from some value to the left of -2, as we get closer to -2, the value of x decreases. As the x-value decreases, the function evaluate \(f(x)=2\left|\frac{x}{x^{2}-4}\right|\) could yield a positive value when close to -2 from left, causing the graph to climb towards infinity. Hence, the limit of \(f(x)\) as \(x\) approaches \(-2\) from the left is \(\infty\).

Step 3: Checking the limit as x approaches -2 from the right

Starting from some value to the right of -2, as we get closer to -2, the x-value decreases. As the x-value decreases, the function evaluate \(f(x)=2\left|\frac{x}{x^{2}-4}\right|\) could yield a positive value when near -2 from the right, causing the function to climb towards infinity. Hence, the limit of \(f(x)\) as \(x\) approaches \(-2\) from the right is also \(\infty\).

Key Concepts

Limits of a Function

Understanding the concept of a limit is essential in calculus. The limit of a function describes the behavior of the function as its argument (commonly represented by the variable \(x\)) gets closer to a certain value. This does not necessarily mean that the function will ever reach that value, but rather it tells us what value the function is approaching.

When analyzing a function for its limit, we ask the question: What value does \(f(x)\) approach as \(x\) gets arbitrarily close to a point? In the context of our exercise, we are interested in the value that \(f(x)\) approaches as \(x\) gets infinitely close to -2, both from the left and the right sides. Limits can approach a real number, or they may be unbounded, approaching infinity (\(\infty\)) or negative infinity (\(-\infty\)).

The function \(f(x)\) in the exercise is particularly interesting because it involves the absolute value, which impacts how it behaves around the point \(x = -2\). As determined through the solution steps, the limit of \(f(x)\) as \(x\) approaches -2 is going towards infinity, regardless of the direction we approach from. This is an illustration of infinite limits, where the values of \(f(x)\) increase without bound as \(x\) approaches a specific point.

Absolute Value in Limits

The absolute value function, denoted as \(|x|\), plays a pivotal role in understanding limits of certain types of functions. It returns the non-negative value of \(x\), meaning it measures the distance of \(x\) from zero on the number line without considering the direction. In terms of limits, this means that when the variable within the absolute value sign approaches a limit point, the function itself will approach the positive value of that limit.

In the given exercise, the function \(f(x) = 2\left|\frac{x}{x^2-4}\right|\) possesses an absolute value in its definition. This assures that the output of the function is non-negative, regardless of the input \(x\) being positive or negative. When inspecting the behavior as \(x\) approaches -2, we must consider that the absolute value will influence the function to reach towards positive infinity – which is the non-negative direction of the outputs. This characteristic of the absolute value within the limit is a turning point for identifying that as \(x\) approaches -2, \(f(x)\) shoots upwards to infinity.

Approaching Infinity

In calculus, when we talk about a function \(f(x)\) approaching infinity, we're describing a scenario where the function's value increases without any upper bound as \(x\) nears a particular point. However, 'approaching infinity' isn't about reaching an actual value; infinity is a concept, not a number.

Infinite limits are a significant part of understanding the behavior of functions, especially when they aren't bounded. As shown in the original exercise, when analyzing the limit of the function \(f(x)\) as \(x\) approaches a specific value, in this case, -2, from either the left or the right, both situations lead to the function's value heading towards infinity. This is indicative of a vertical asymptote at \(x = -2\) where the graph of the function would visually display a line shooting upwards without end. Approaching infinity can have profound implications, such as in cases of finding horizontal asymptotes or determining the growth rate of functions, which are foundational concepts in the study of calculus.