Question

In Exercises \(35-38\), use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l} f(x)=\sec \frac{\pi x}{6} \\ \lim _{x \rightarrow 3+} f(x) \end{array} $$

Short answer

The exact value of the limit would depend on the graph. However, given that \(\sec x = 1/ \cos x\), we know that the function will be undefined wherever cosine is 0. For the cosine function, this is every point of \( \frac{(2k+1)\pi}{2} \), where \(k\) is an integer. If \( \frac{\pi x}{6} \) matches this form when \( x = 3 \), the function will have a vertical asymptote at this point and the limit from the right will be \(-\infty\). Otherwise, the limit will be \( \sec( \frac{\pi x}{6} ) \) calculated at the point just greater than \(3\).

Step-by-step solution

Understanding the function

The function given is \(f(x)=\sec \frac{\pi x}{6}\). The sec is a reciprocal of cosine function, which means \(\sec x = 1/ \cos x\). It is undefined for values of x where cosine is 0. In this case, we need to determine its behavior near the point \(x=3\). Specifically, we want to know the value of the function as x is slightly more than 3, represented mathematically as \(x \rightarrow 3+\).

Graphing the function

Graph the function \(f(x)=\sec \frac{\pi x}{6}\) using a graphing utility, marking the point \(x = 3\) specifically. Pay special attention to the behavior of the function near this point. The function will have vertical asymptotes at points where the value is undefined.

Evaluating the limit

The one-sided limit \( \lim _{x \rightarrow 3+} f(x)\) represents the value that the function approaches as \(x\) approaches \(3\) from greater than \(3\). Inspect this behavior on the graph. If the function value is tending towards a specific number, that is the limit. If it is tending towards infinity (\(+\infty\) or \(-\infty\)), that is the limit. If it does not tend towards a specific value, the limit does not exist.

Key concepts

Secant Function

When studying trigonometric functions, it's important to understand the secant function, often denoted as \(sec x\). The secant function is the reciprocal of the cosine function, which means \(sec x = \frac{1}{\cos x}\). Its graph is not continuous and has distinctive patterns that are crucial for grasping the concept of limits and asymptotic behavior.
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Graphically, the secant function consists of an infinite number of branches, each occurring between the vertical lines where the cosine function equals zero. These vertical lines are where the secant function has vertical asymptotes, as the value of secant approaches infinity. As with all trigonometric functions, understanding the secant function requires familiarity with the unit circle and the behavior of the associated cosine values.

Graphing Utilities

Graphing utilities, such as graphing calculators or computer software, are invaluable tools when it comes to visualizing mathematical functions. For trigonometric functions like the secant function, they allow us to see the curve's behavior over a range of values, identify points of interest such as asymptotes and intercepts, and determine end behavior or limits.

Using graphing utilities, students can dynamically manipulate the view of a function's graph; zooming in and out provides a better understanding of the function's local and global behavior. It is a particularly useful method for observing the immediate behavior of functions as they approach specific values, which is essential when evaluating one-sided limits.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus, used to find the value that a function approaches as the input approaches a certain point. It is crucial for understanding continuous and discontinuous points on a graph. In the context of the secant function \(f(x) =\sec \frac{\pi x}{6}\), evaluating a one-sided limit, such as \(\lim _{x \rightarrow 3+} f(x)\), involves observing the function's value as \(x\) approaches 3 from the right (i.e., from values greater than 3).

If the function approaches a particular value, that is the limit. However, if the function increases or decreases without bound or fails to settle at any one value, the limit may be infinity or not exist at all. Evaluating one-sided limits often involves close observation of a graph near the point of interest, which is where graphing utilities become particularly useful.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. These can be vertical, horizontal, or oblique (slant). In the case of secant and other trigonometric functions, vertical asymptotes are particularly important. They occur at input values where the function becomes undefined - for example, where the cosine value is zero, and, consequently, its reciprocal, the secant, tends to infinity.

Vertical asymptotes represent a kind of 'boundary' at which a function's value becomes infinitely large or small in magnitude. Identifying vertical asymptotes is a crucial step in graphing the function correctly, as well as in understanding its limit behavior near those boundaries. When evaluating one-sided limits, if the point of interest is at an asymptote, the limit typically will either approach infinity, negative infinity, or it may not exist depending on the direction from which the value is approached.