Question

Evaluate the given limit. $$ \lim _{t \rightarrow 0}\left\langle\frac{t}{\sin t},(1+t)^{\frac{1}{t}}\right\rangle $$

Short answer

The limit is \( \langle 1, e \rangle \).

Step-by-step solution

Evaluate limit for the first component

The first component of the vector is \( \frac{t}{\sin t} \). We apply the standard limit \( \lim_{t \to 0} \frac{t}{\sin t} = 1 \). Therefore, the limit for the first component is 1.

Evaluate limit for the second component

The second component is \( (1+t)^{\frac{1}{t}} \). This is a common form of the exponential limit \( e^x = \lim_{t \to 0} (1+x)^{\frac{1}{x}} \). Therefore, the limit of the second component as \( t \to 0 \) is \( e \).

Combine results

The vector limit is formed by combining the results of the evaluated limits of each component. As \( t \to 0 \), the limit of the vector is \( \langle 1, e \rangle \).

Key concepts

Understanding Vector Calculus in Limits

Vector calculus allows us to extend the concepts of calculus like limits to multiple dimensions. In our exercise, the vector is composed of two components: \( \frac{t}{\sin t} \) and \( (1+t)^{\frac{1}{t}} \). Each of these components behaves like a scalar function, but together, they form a vector in a mathematical space.
Evaluating the limit of a vector involves considering each component separately and then combining their individual results. It's critical to treat each component independently because they may approach their limits at different rates or even to different values. This approach is common in vector calculus, helping us analyze multi-variable functions effectively. By breaking down complex problems into more manageable parts, we gain better insight and arrive at precise conclusions for seemingly complicated vector expressions.

Trigonometric Limits Illustrated

Trigonometric limits often appear in calculus, particularly when evaluating limits that involve angles and their sines, cosines, or other trigonometric functions. In the exercise, we're looking at \( \lim_{t \to 0} \frac{t}{\sin t} = 1 \). This is a classic trigonometric limit derived from the fact that as the angle approaches zero, the sinusoidal value is approximately equal to the angle's measure in radians.
Understanding this limit is critical because it frequently appears in calculus problems, especially involving small-angle approximations. It's useful in fields from physics to engineering, providing a simplified method to approach complex trigonometric functions. By remembering that \( \frac{t}{\sin t} \) tends to 1 as \( t \) approaches 0, you can solve a wide array of related problems using this foundational concept.

Exploring Exponential Limits

Exponential limits offer a fascinating glimpse into how functions behave when subjected to infinite processes. The limit \( \lim_{t \to 0}(1+t)^{\frac{1}{t}} = e \) is the foundation of Euler's number \( e \), a fundamental constant in mathematics.
This form often appears when dealing with compound interest problems, continuous growth phenomena, and certain calculus problems involving exponential functions. The idea is that as \( t \) shrinks towards zero, the expression\( (1+t)^{\frac{1}{t}} \) approaches the transcendental number \( e \), which is approximately equal to 2.71828. This concept not only provides insights into natural growth processes but also plays a crucial role in logarithmic calculations and more complex algebraic manipulations.
By understanding how exponential functions behave in the limit, you can tackle a broad range of mathematical problems more confidently and derive deeper insights into the nature of continuous growth and decay.