Chapter 2: Problem 38
Determine the following limits.
Short Answer
Expert verified
Answer: The limit of the function f(θ) as theta approaches pi/2 from the positive side is .
Step by step solution
01
Identify the function and its limits
In this case, the function we are working with is The limit we need to determine is where the notation indicates that theta approaches pi/2 from the positive side, or slightly greater than pi/2.
02
Determine the behavior of the trigonometric function near pi/2
As theta approaches pi/2 from the positive side, i.e., when theta is slightly greater than pi/2, the value of increases towards +∞. This is because the tangent function has a vertical asymptote at .
03
Calculate the limit of the function
Given the behavior of the tangent function near pi/2, we can rewrite the limit expression as:
Since the tangent function increases towards +∞ as theta approaches pi/2 from the positive side, we have:
Therefore, the limit expression becomes:
.
04
Express the final answer
Since the infinite limit is scaled by a constant factor 1/3, the limit still remains infinite:
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Calculus
Calculus, the mathematical study of continuous change, is a branch that deals extensively with functions, derivatives, integrals, and limits. The concept of a limit is fundamental in calculus because it underpins the definition of derivatives and integrals. It involves understanding the behavior of a function as the input approaches a particular value. This is especially crucial when the function behaves erratically or tends to infinity near that point, which is often the case with trigonometric functions near certain angles.
In the context of the given exercise, calculus provides us with tools to deal with the intricate behavior of the tangent function as it approaches its vertical asymptote. By analyzing the limit of a function, we can predict how it behaves near points of discontinuity or where it's undefined, allowing us to handle complex mathematical phenomena in a rigorous fashion.
In the context of the given exercise, calculus provides us with tools to deal with the intricate behavior of the tangent function as it approaches its vertical asymptote. By analyzing the limit of a function, we can predict how it behaves near points of discontinuity or where it's undefined, allowing us to handle complex mathematical phenomena in a rigorous fashion.
Tangent Function Behavior
The tangent function, denoted as , is a periodic function that shows up frequently in trigonometry and calculus. Its behavior is characterized by a repeating pattern of upward and downward spikes, resulting from its definition as the ratio of the sine and cosine functions. A key feature of the tangent function is the presence of vertical asymptotes—lines that the function approaches but never touches or crosses.
In particular, at each odd multiple of , the tangent function has a vertical asymptote. This means as the input approaches an odd multiple of from either side, the value of the tangent function shoots off to positive or negative infinity. Understanding this behavior helps in analyzing limits that involve the tangent function, as we are tasked with in the given exercise.
In particular, at each odd multiple of
Limit of a Function
The limit of a function is a fundamental concept that describes the value that a function approaches as the input approaches a certain point. Limits are essential for analyzing the behavior of functions, especially at points where they are not well-defined or where they exhibit erratic behavior (such as division by zero or infinity).
In formal terms, if approaches a certain value as approaches a value from either direction, we write this as . If the function grows without bound, we say the limit is infinity, expressed symbolically as . The proper understanding of a function's limit not only solidifies one's grasp of continuous functions but also prepares one for more advanced calculus topics like derivatives and integrals.
In formal terms, if
Approaching Vertical Asymptotes
When dealing with functions that involve division, like ratios of polynomials or trigonometric functions, we often encounter vertical asymptotes. These are vertical lines at certain x-values where the function does not have a finite limit because the values of the function increase or decrease without bound as the input approaches the asymptote's x-value.
As an input value gets closer to the vertical asymptote, the function's output will grow towards positive or negative infinity, depending on the direction from which the input approaches. For instance, when dealing with the tangent function, we typically notice that as the angle approaches from the right, denoted as , the output heads towards positive infinity. This characterizes the 'infinity' behavior of limits related to vertical asymptotes and is essential for solving problems involving the limits of functions that have these asymptotes.
As an input value gets closer to the vertical asymptote, the function's output will grow towards positive or negative infinity, depending on the direction from which the input approaches. For instance, when dealing with the tangent function, we typically notice that as the angle approaches