Problem 10 - Calculus III

Question

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow \infty}\left\langle e^{-2 t}, \frac{2 t+3}{3 t-1}, \arctan (2 t)\right\rangle $$

Step-by-Step Solution

Answer

The limit is \(\langle 0, \frac{2}{3}, \frac{\pi}{2} \rangle.\)

Step 1: Determine the Limit of First Component

The first component of the vector-valued function is \(e^{-2t}\). As \(t \rightarrow \infty\), the function \(e^{-2t}\) approaches zero since exponentials with negative exponents decay to zero as the exponent approaches infinity. Thus, \(\lim_{t \to \infty} e^{-2t} = 0\).

Step 2: Determine the Limit of the Second Component

The second component of the vector-valued function is \(\frac{2t+3}{3t-1}\). To find the limit as \(t \rightarrow \infty\), divide the numerator and denominator by \(t\): \(\frac{2 + \frac{3}{t}}{3 - \frac{1}{t}}\). As \(t \rightarrow \infty\), the terms \(\frac{3}{t}\) and \(\frac{1}{t}\) approach zero, leaving \(\frac{2}{3}\). Thus, \(\lim_{t \to \infty} \frac{2t+3}{3t-1} = \frac{2}{3}\).

Step 3: Determine the Limit of the Third Component

The third component is \(\arctan(2t)\). As \(t \rightarrow \infty\), \(2t\) also approaches infinity, and so \(\arctan(2t)\) approaches \(\frac{\pi}{2}\) since the arctangent of a very large positive number approaches \(\frac{\pi}{2}\). Therefore, \(\lim_{t \to \infty} \arctan(2t) = \frac{\pi}{2}\).

Step 4: Compile Results into the Limit of the Vector

Combine the limits of each component found in the previous steps to determine the limit of the entire vector-valued function. Thus, \[\lim_{t \to \infty}\langle e^{-2t}, \frac{2t+3}{3t-1}, \arctan(2t) \rangle = \langle 0, \frac{2}{3}, \frac{\pi}{2} \rangle.\]

Key Concepts

Vector-Valued Functions

Vector-valued functions are a type of mathematical expression where instead of producing a single output, you produce a vector. A vector is essentially a list of values, called components. Each component function operates independently, but they are all related by the variable they share. In multivariable calculus, vector-valued functions often depend on variables like time, denoted as \(t\), which is commonly used in physics to model motion.

• Each component in a vector-valued function could be a different type of function, like polynomials, exponentials, or trigonometric functions.
• These functions are often written in angle brackets, such as \(\langle f(t), g(t), h(t) \rangle\).
• When finding limits of vector-valued functions, you evaluate the limit of each component individually.

Understanding these functions allows us to tackle complex real-world problems where multiple interrelated quantities change over time.

Limits

Limits are a fundamental concept in calculus used to describe how a function behaves as its input approaches a particular value. In the context of a vector-valued function, finding the limit means considering the limits of each individual component separately.

• For each component of the vector, examine what value it approaches as the variable \(t\) approaches a certain point, such as infinity.
• The limit gives insight into the behavior of complex functions and helps describe continuity, differentiability, and integrability of functions.
• When evaluating limits, simplifying the expression as much as possible can often reveal insights into the behavior of the function.

By understanding limits, we can predict the behavior of functions and create more meaningful models for dynamic systems.

Exponential Functions

Exponential functions are mathematical expressions in which variables appear as exponents. They are crucial in many scientific areas for modeling growth and decay processes. When dealing with limits, especially when the exponent is negative, like in \(e^{-2t}\), the behavior of the exponential function becomes very significant.

• For \(e^{-2t}\), as \(t\) approaches infinity, the function value approaches zero. This behavior highlights how exponential decay works.
• Exponential functions with positive exponents grow rapidly, whereas those with negative exponents diminish toward zero quickly.
• They define many natural phenomena, like population growth or radioactive decay.

Exponential functions provide a great tool to describe processes that change rapidly with time or space, and their limits help us understand these changes in a long-term context.

Arctangent Function

The arctangent function, represented as \(\arctan(x)\), is the inverse of the tangent function. It is an essential function in trigonometry, helping to find angles when given the tangent ratio. The behavior of arctangent is particularly interesting at extremes, such as when approaching infinity.

• The function \(\arctan(x)\) approaches \(\frac{\pi}{2}\) as \(x\) approaches infinity.
• It is important in calculations involving angles and can help determine direction and orientation in vector analyses.
• The arc functions like \(\arctan\) are bounded, meaning they never exceed certain values, which makes them useful in handling real-world phenomena where quantities naturally have upper bounds.

Understanding the behavior of the arctangent function at limits helps in comprehending more complex functions that can incorporate angle measures or changes over time.