Question

In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow \infty} e^{x} \sin e^{-x^{2}} $$

Short answer

Answer: The limit of the function \(e^x \sin e^{-x^2}\) as x approaches infinity is 0.

Step-by-step solution

Find the limit of the exponential function

As x approaches infinity, we know that the exponential function \(e^x\) also approaches infinity. Therefore: $$ \lim _{x \rightarrow \infty} e^{x} = \infty $$

Find the limit of the trigonometric function

To find the limit of the trigonometric function, we first need to find the limit of the term inside the sine function, which is \(e^{-x^2}\). As x approaches infinity, the term inside the exponent becomes increasingly large and negative, so the value of the exponential function approaches zero: $$ \lim_{x \rightarrow \infty} e^{-x^2} = 0 $$ Now, using this result, we can find the limit of the trigonometric function: $$ \lim_{x \rightarrow \infty} \sin e^{-x^2} = \lim_{x \rightarrow \infty} \sin 0 = 0 $$

Combine the limits of two functions

Now that we have the limits of the two functions as x approaches infinity, we can multiply them to find the overall limit of the given function: $$ \lim_{x \rightarrow \infty} e^x \sin e^{-x^2} = \left(\lim_{x \rightarrow \infty} e^x\right)\left(\lim_{x \rightarrow \infty} \sin e^{-x^2}\right) = \infty \cdot 0 $$

Evaluate the product of limits

The product of \(\infty\) and \(0\) is an indeterminate form. However, in this case, we can use the Squeeze Theorem to evaluate the limit. We know that the sine function has a range of \(-1 \leq \sin x \leq 1\), so: $$ -1 \leq \sin e^{-x^2} \leq 1 $$ Multiplying by \(e^x\) (which is always positive as x approaches infinity), we get: $$ -e^x \leq e^{x} \sin e^{-x^2} \leq e^x $$ Now, as x approaches infinity, we know that \(e^x\) approaches infinity and \(-e^x\) approaches negative infinity. However, since \(\sin e^{-x^2}\) is bounded between -1 and 1, the overall limit of the function will be: $$ \lim_{x \rightarrow \infty} e^{x} \sin e^{-x^2} = 0 $$ So, the limit of the given function as x approaches infinity is 0.

Key concepts

Limits and Continuity

Understanding the concept of limits is crucial when studying calculus, as it deals with behavior of functions as they approach a certain point or infinity. Specifically, continuity implies that a function is smooth and uninterrupted within its domain. When we talk about the limit of a function as it approaches infinity, like in the problem \( \lim _{x \rightarrow \infty} e^{x} \sin e^{-x^{2}} \), we are interested in the behavior of the function for very large values of x.

The concept of continuity is important in limits because, for a function to have a limit as x approaches a particular value, the function must approach a single finite number. If we find that a function does not converge to a specific value, it means the limit does not exist. However, if a function approaches a specific value, even at infinity, it's said to have a limit at that point.

If a limit leads to an indeterminate form, such as \(\infty \cdot 0\), additional techniques like the Squeeze Theorem are employed to find the limit. The Squeeze Theorem is especially useful because it confines the function between two other functions that have known limits at a certain point.

Exponential Functions

An exponential function is a mathematical expression in which a constant base is raised to a variable power. An important feature of exponential functions, like \(e^x\), is that their rate of growth increases exponentially as x becomes larger, thereby having profound implications in various scientific fields including biology, finance, and physics.

In the given exercise, we see the exponential function \(e^x\) as x approaches infinity. Intuitively, as x grows larger, the expression \(e^x\) increases without bound, approaching infinity. This behavior is symbolic of all exponential growth patterns—they don't just increase; they do so at an accelerating pace.

When calculus problems involve limits of exponential functions, they often test the understanding of this accelerating growth, especially as x approaches very large numbers, or conversely, very negative numbers, where the function approaches zero, such as in \(e^{-x^2}\) as x approaches infinity.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are fundamental in math because they relate the angles of triangles to the lengths of their sides. With a deep roots in geometry, these functions have expanded their utility to modeling periodic phenomena in many scientific fields.

The interesting property of the sine function, denoted as \(\sin x\), is that it oscillates between -1 and 1. This means no matter what the input is, the output is restricted to this range. This property is utilized in the exercise where we evaluate \(\sin e^{-x^2}\) as x approaches infinity. Since the exponent \(e^{-x^2}\) approaches zero, \(\sin e^{-x^2}\) approaches \(\sin 0\), which is also zero.

Because trigonometric functions are periodic and bounded, they introduce a unique behavior when combined with unbounded functions like exponential functions, leading to solutions that hinge upon understanding the limits and continuity of both types of functions working together.