Question

Multiple Choice \(\lim _{x \rightarrow 2^{-}} \frac{x}{x-2}\) \((A)-\infty \quad(B) \infty\) (C) 1 (D) \(-1 / 2 \quad(\mathrm{E})-1\)

Short answer

The limit is \(-\infty\). Hence, from the options (A) - \(-\infty\) is the correct answer.

Step-by-step solution

Identify what the limit is asking

In this step, understand that the limit \(x \rightarrow 2^{-}\) means \(x\) is approaching \(2\) from the left side. This implies that \(x\) is marginally less than \(2\).

Identify what values \(x\) can take

Given \(x\) approaches \(2\) from the left, \(x\) is slightly less than \(2\). Consider \(x = 1.99999\), a value slightly lesser than \(2\). Substitute \(1.99999\) into the function: \(\frac{1.99999}{1.99999-2}\)

Simplify the equation

After substituting \(x\) into the function, we get \(\frac{1.99999}{-0.00001}\). Upon simplification, we arrive at \(-199999\), which is a very large negative number. As \(x\) gets closer and closer to \(2\) from the left, this value tends directly to \(-\infty\).

Key concepts

Limits in Calculus

Limits in calculus are fundamental to understanding the behavior of functions as inputs approach certain values. They help us grasp what happens to function outputs when inputs get very close to some specific point but don't actually reach it.

When we see the notation \( \lim_{x \rightarrow a} f(x) \), it is an instruction to examine what value the function \(f(x)\) is getting closer to as \(x\) approaches \(a\). This could be a finite number or, in some cases, infinity or negative infinity. When a limit exists and is finite, we can often use direct substitution to find it. However, when we're dealing with forms that are undefined, like \(\frac{1}{0}\), other methods, such as factoring, rationalizing, or looking at the graph, need to be employed.

It's essential to approach limits analytically and with a clear understanding of function behavior, especially near points of discontinuity. This knowledge will enable students to better understand calculus and how it applies to real-world situations where gradual change is observed.

Asymptotic Behavior

When discussing limits and functions, the concept of asymptotic behavior is frequently mentioned. This refers to how a function behaves as it moves closer to a given line or value without actually reaching it. An 'asymptote' is this line or value that the function approaches but does not touch or cross.

We can have vertical asymptotes, which occur when the value of a function becomes unbounded as the input approaches a certain point, or horizontal asymptotes, which define the behavior of a function as the input goes to positive or negative infinity. The exercise in question presents a situation where the function \(f(x)=\frac{x}{x-2}\) has a vertical asymptote at \(x=2\), which means as \(x\) approaches 2 from the left, the function values tend towards negative infinity. This observation is a vital characteristic of asymptotic behavior and is crucial for understanding the limits of functions with discontinuities.

Approaching Infinity

The concept of approaching infinity in calculus refers to when a variable grows without bound, either positively or negatively, as it nears a particular point. It’s an abstract idea, associated with things that are not finite.

In the context of the exercise provided, \( x \), which is approaching \( 2 \) from the left side, means that we have to consider values of \( x \) that are just a bit smaller than \( 2 \) and see what happens to our function \( f(x) \). As \( x \) gets closer and closer to \( 2 \), specifically from the left (\( 2^{-} \)), the result of the function grows negatively larger, indicating the function heads towards negative infinity. This notion of infinity is a non-tangible, yet powerful tool used by mathematicians to describe scenarios that, although they can’t technically exist in a physical sense, provide insight into the behavior of functions in extreme cases.

Unbounded Behavior

Unbounded behavior in a mathematical sense describes functions that don't have a finite limit; their output values can grow indefinitely. This term is often used to describe functions that have vertical asymptotes, where the function approaches a certain input value and the output either increases or decreases without limit.

Within our exercise, we analyze a function that displays unbounded behavior as \( x \) approaches \( 2 \) from the left (\( 2^{-} \)). It's because the denominator of our function, \( x-2 \) becomes very small, approaching zero, which results in the function's value becoming a very large negative number. As \( x \) continues to get closer to \( 2 \) from the left, the output value will also continue to decrease indefinitely, hence the function is considered to have unbounded behavior at that point.