Question

One-Sided Limit In Exercises \(49-52\) , use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l}{f(x)=\frac{x^{2}+x+1}{x^{3}-1}} \\ {\lim _{x \rightarrow 1^{+}} f(x)}\end{array} $$

Short answer

To fully answer this question, one would need to calculate the limit as per Step 3. This depends on what is observed in the graph from Step 1. If no clear value or anomaly is observed from the graph, it is likely that the function approaches a specific, calculable value as x approaches 1 from the right, which would be the output of the calculated limit.

Step-by-step solution

Graph the function using a graphing utility

Enter the function \(f(x)=\frac{x^{2}+x+1}{x^{3}-1}\) into a graphing utility. Make sure to narrow your window around x = 1 to get a detailed view of what happens to the function as it nears 1.

Analyze the graph

From the graph, observe if the function is approaching a specific value as x approaches 1 from the right. If you notice any jumps, discontinuities, or asymptotes near x = 1, make a note of them as they will impact our limit calculation.

Calculate the limit as x approaches 1 from the right

Substitute x = 1 into the given function. That is, calculate the value of \(f(x)=\frac{x^{2}+x+1}{x^{3}-1}\) for \(x=1\) and report. Remember since we are looking for the limit as x approaches 1 from the right, we are interested in values of x that are only just greater than 1.

Key concepts

Limits in Calculus

Understanding limits is a cornerstone of calculus. In essence, the limit describes the value that a function approaches as the input (or 'x') gets closer to a certain point. One-sided limits are a more focused type of limit; they tell us what happens to a function when we approach a specific point from one side only. This can be from the right (denoted as \( x \rightarrow c^{+} \)) or from the left (denoted as \( x \rightarrow c^{-} \)). They are particularly useful in understanding the behavior of functions that are not well-behaved at some points—points where the function doesn’t have a certain value for an \( x \) value, or points where the function value becomes infinite, known as discontinuities or asymptotes. When using a graphing utility, observing the trend of the curve as it approaches the point can provide valuable insight into one-sided limits.

Graphing Functions

Graphing functions plays a significant role in visualizing the behavior of equations. When it comes to calculus, a graph can communicate a lot of information quickly, revealing where a function increases or decreases, as well as its continuity and differentiability. When graphing for the purpose of determining one-sided limits, it's crucial to zoom in on the area of interest and identify patterns as the function nears the particular x-value from one side. This graphical analysis can show subtleties not always clear from the formula alone, such as hidden asymptotes or unexpected behavior in the function's output.

It’s essential to use graphing utilities accurately by inputting functions correctly and setting appropriate scales for the axes. A well-drawn graph provides a visual context that can be invaluable for predicting and verifying the limits of a function.

Asymptotes and Discontinuities

Asymptotes are lines that a graph approaches but never actually touches or crosses. They represent the behavior of functions whose values grow without bound (to infinity) or shrink towards an infinitely small point (to zero), as they get closer to a certain x-value. Discontinuities, on the other hand, occur in places where a function 'jumps' or has a break, such as in the case of a piecewise function or when there's a division by zero.

Identifying these aspects in a graph is crucial, especially when dealing with one-sided limits. If a function approaches a different value as \( x \rightarrow c^{+} \) than it does as \( x \rightarrow c^{-} \) or does not approach a value at all, the limit at \( x = c \) does not exist. However, the one-sided limits may still exist, showing us the behavior of the function from each individual direction.

Calculus Graphical Analysis

Calculus graphical analysis involves interpreting the features of function graphs to understand the underlying behavior of different calculus concepts, such as limits, derivatives, and integrals. For instance, by examining the slope of a tangent to a function's graph at different points, we can infer the function's rate of change, which is the derivative. Similarly, when analyzing limits, the value a function graph tends towards as the input approaches a particular value can give us insight into the limit.

In the context of the exercise provided, the one-sided limit as \( x \rightarrow 1^{+} \) for the function \( f(x)=\frac{x^{2}+x+1}{x^{3}-1} \) can be analyzed by closely inspecting the graph near \( x = 1 \). Even if a function is undefined at the point, as in a division by zero, the graph can show us how the function behaves near that point, offering clues on the one-sided limit, which is vital for fully grasping the function's nature around discontinuities and asymptotes.