Question

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 2^{+}} \frac{x}{x-2} $$

Short answer

The one-sided limit of \( \frac{x}{x-2} \) as \(x\) approaches 2 from the right is positive infinity

Step-by-step solution

Understand the function

The given function is \( \frac{x}{x-2} \). It's a simple rational function.

Identify behavior as x approaches 2 from the right

As \(x\) approaches 2 from the right (i.e., from values slightly greater than 2), the denominator \(x-2\) becomes very small and positive since we are subtracting a number very close to \(x\). Therefore, our fraction \( \frac{x}{x-2} \) becomes increasingly large and positive, which tends to positive infinity.

Provide the answer to the one-sided limit

Therefore, the limit of this function as \(x\) approaches 2 from the right is positive infinity.

Key concepts

Limits and Continuity

In calculus, understanding limits is fundamental to exploring how functions behave near a certain point. Specifically, a limit attempts to describe the value that a function approaches as the input (or 'x' value) gets close to some number. For instance, when examining the limit as x approaches 2 from the right for the function \( \frac{x}{x-2} \), we ask ourselves: What value does \( \frac{x}{x-2} \) approach as x gets arbitrarily close to 2, from values greater than 2?

Continuity, on the other hand, involves whether a function has any breaks, jumps, or holes at a certain point. A function is continuous at a point if the limit exists at that point, and the function’s value equals that limit. In the case of our given function, \( \frac{x}{x-2} \) is not continuous at x = 2 since the function becomes undefined due to a division by zero as x is exactly 2. However, understanding the concept of limits, specifically one-sided limits, still gives us insight into the behavior of functions near points of discontinuity.

Rational Functions

Rational functions are fractions involving polynomials. In the exercise, \( \frac{x}{x-2} \) is a rational function with a numerator of 1 (the polynomial 'x') and a denominator of \( x-2 \) (a polynomial of 'x'). The intriguing part of rational functions comes into play near values that make the denominator zero; these values are termed 'singularities.'

At x = 2, we have a singularity, as the denominator \( x-2 \) equals zero, making the function undefined. When analyzing limits near these singular points, rational functions can exhibit varied behavior - shooting off to infinity, as in our case, or oscillating between extremes. It’s these points that often necessitate the study of one-sided limits to fully grasp how the function behaves as we approach from one direction only.

Infinite Limits

In certain cases, as with the exercise function \( \frac{x}{x-2} \), as x approaches a particular value, the output of the function grows without bound. This is termed an 'infinite limit.' In our function, as x gets closer to 2 from the right, the denominator becomes very small and positive, which causes the value of the function to increase without limit.

It’s essential to note that an infinite limit doesn't mean the function will equal infinity at the singularity. Instead, it indicates that there is no real number that can be described as the limit. Such behavior is crucial in calculus as it helps identify asymptotic behavior where the function grows closer to a line (the vertical asymptote) without ever touching it as we zoom in on the graph around that singular point.