Problem 19
Question
Find the limit. $$ \lim _{t \rightarrow-(3 / 2)^{-}} \sec \pi t $$
Step-by-Step Solution
Verified Answer
The limit of the given function, \(\lim_{t \rightarrow (-\frac{3}{2})^{-}} \sec \pi t\), does not exist because the secant function approaches a vertical asymptote at this point.
1Step 1: Understand the secant function
The secant function is defined as the reciprocal of the cosine function, that is:
\[
\sec x = \frac{1}{\cos x}
\]
We also know that the cosine function has a period of \(2\pi\) and has zeros at odd multiples of \(\frac{\pi}{2}\), which means the secant function will have vertical asymptotes (i.e., undefined) at those points.
2Step 2: Evaluate the inner expression
In this limit, we have \(πt\) inside the secant function. We first need to evaluate this expression as \(t\) approaches \(\frac{-3}{2}\) from the left side:
\[
\pi t = \pi \left(-\frac{3}{2}\right) = -\frac{3\pi}{2}
\]
3Step 3: Analyze the secant function at the given point
Now we need to consider the behavior of the secant function as the angle inside approaches \(\frac{-3 \pi}{2}\). From Step 1, we know that the secant function has vertical asymptotes at odd multiples of \(\frac{\pi}{2}\), and since \(\frac{-3\pi}{2}\) is an odd multiple of \(\frac{\pi}{2}\), the secant function will have a vertical asymptote at this point.
4Step 4: Determine the limit
As \(t\) approaches \(\frac{-3}{2}\) from the left, the argument inside the secant function approaches \(\frac{-3\pi}{2}\) and the secant function approaches a vertical asymptote. Therefore, the limit does not exist, as the function's value becomes unbounded at this point. Thus, we can write
\[
\lim_{t \rightarrow (-\frac{3}{2})^{-}} \sec \pi t = \text{Does not exist}
\]
As a result, the limit of the secant function as \(t\) approaches \(-\frac{3}{2}\) from the left side does not exist due to the presence of a vertical asymptote.
Key Concepts
Secant FunctionVertical AsymptotesLimit Does Not Exist
Secant Function
The secant function, often denoted as \( \sec x \), plays a prominent role in trigonometry and calculus. It is defined as the reciprocal of the cosine function, thus \( \sec x = \frac{1}{\cos x} \) where \(x \) is an angle measured in radians. Since \( \sec x \) is the inverse of \( \cos x \) and cosine has a range of [-1, 1], the secant function has certain restrictions.
It is important to note that whenever the cosine function equals zero, the secant function becomes undefined because division by zero is not possible in mathematics. The values at which \( \cos x = 0 \) are odd multiples of \( \frac{\pi}{2} \) (such as \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), etc.). Consequently, the secant function exhibits interesting behavior at these points, which leads to the creation of vertical asymptotes, a concept we'll explore further in the next section.
In calculus, understanding the secant function and its behavior is essential, especially when evaluating limits, as it can encounter points where the value of the function goes to infinity.
It is important to note that whenever the cosine function equals zero, the secant function becomes undefined because division by zero is not possible in mathematics. The values at which \( \cos x = 0 \) are odd multiples of \( \frac{\pi}{2} \) (such as \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), etc.). Consequently, the secant function exhibits interesting behavior at these points, which leads to the creation of vertical asymptotes, a concept we'll explore further in the next section.
In calculus, understanding the secant function and its behavior is essential, especially when evaluating limits, as it can encounter points where the value of the function goes to infinity.
Vertical Asymptotes
When dealing with functions, the term 'vertical asymptote' refers to a line (or lines) that a function approaches but never actually reaches or crosses. In a graphical representation, the value of the function either increases or decreases without bound as it gets closer to the line of the vertical asymptote.
For the secant function, as we learned earlier, vertical asymptotes occur at odd multiples of \( \frac{\pi}{2} \). The concept of vertical asymptotes is key in identifying behavior of transcendental functions (functions like sine, cosine, secant, etc.) and especially in determining the existence of limits at certain points.
When a function approaches a vertical asymptote, we say that the limit of the function does not exist (DNE) at that point because the function is not approaching a finite value but instead is heading towards positive or negative infinity, depending on the direction of the approach. The presence of a vertical asymptote is often a signal that the function will see unbounded behavior at certain specific inputs.
For the secant function, as we learned earlier, vertical asymptotes occur at odd multiples of \( \frac{\pi}{2} \). The concept of vertical asymptotes is key in identifying behavior of transcendental functions (functions like sine, cosine, secant, etc.) and especially in determining the existence of limits at certain points.
When a function approaches a vertical asymptote, we say that the limit of the function does not exist (DNE) at that point because the function is not approaching a finite value but instead is heading towards positive or negative infinity, depending on the direction of the approach. The presence of a vertical asymptote is often a signal that the function will see unbounded behavior at certain specific inputs.
Limit Does Not Exist
In calculus, when we mention that a limit does not exist, we are indicating that there is no single finite value that the function approaches as the input approaches a certain value. There can be several reasons for a limit not existing, with one being the presence of a vertical asymptote, as discussed previously.
For the secant function, as the angle approaches an odd multiple of \( \frac{\pi}{2} \) from either direction, the function grows without bound or decreases without bound, therefore, the limit as we approach these points does not exist.
In practice, when calculating limits, if you encounter a situation where an asymptote is present, it is a strong indication that you will end up with a result where the limit does not exist. This understanding is key to successfully solving many calculus exercises involving limits and will undoubtedly assist in predicting and explaining the behavior of functions around these critical points.
For the secant function, as the angle approaches an odd multiple of \( \frac{\pi}{2} \) from either direction, the function grows without bound or decreases without bound, therefore, the limit as we approach these points does not exist.
In practice, when calculating limits, if you encounter a situation where an asymptote is present, it is a strong indication that you will end up with a result where the limit does not exist. This understanding is key to successfully solving many calculus exercises involving limits and will undoubtedly assist in predicting and explaining the behavior of functions around these critical points.
Other exercises in this chapter
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