Question

Determine the following limits. $$\lim _{x \rightarrow \infty} \frac{\sin x}{e^{x}}$$

Short answer

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

Step-by-step solution

Identify the form of the limit

Determine the form of the limit as x approaches infinity: $$\lim_{x \rightarrow \infty} \frac{\sin x}{e^x}$$ As x approaches infinity, the sine function oscillates between -1 and 1, whereas the exponential function e^x grows without bound. This limit is of the form \(\frac{0}{\infty}\).

Apply L'Hôpital's Rule

Since the limit is of the form \(\frac{0}{\infty}\), we can apply L'Hôpital's Rule. This rule states that if \[\lim_{x \rightarrow c} \frac{f(x)}{g(x)}\] is of the form \(\frac{0}{0}\) or \(\frac{\pm \infty}{\pm \infty}\), and both f(x) and g(x) are differentiable functions, then $$\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}$$ In our case, let f(x) = \(\sin x\) and g(x) = \(e^x\). Then, we need to find the derivatives of f(x) and g(x): $$f'(x) = \cos x$$ $$g'(x) = e^x$$ Apply L'Hôpital's Rule and calculate the new limit: $$\lim_{x \rightarrow \infty} \frac{\cos x}{e^x}$$

Evaluate the limit

The new limit is still of the form \(\frac{0}{\infty}\) and we don't need to apply L'Hôpital's Rule again. As x approaches infinity, the cosine function oscillates between -1 and 1, whereas the exponential function e^x continues to grow without bound. As a result, the numerator remains bounded while the denominator increases indefinitely. Thus, the limit converges to 0. So, the limit of the given function is: $$\lim_{x \rightarrow \infty} \frac{\sin x}{e^x} = 0$$

Key concepts

Understanding L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for evaluating limits that present an indeterminate form, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When faced with a limit that is difficult to evaluate using standard algebraic techniques, L'Hôpital's Rule provides a way forward.

To apply L'Hôpital's Rule, both the numerator and denominator of the function must be differentiable. If the limit \( \lim_{x \rightarrow c} \frac{f(x)}{g(x)} \) results in an indeterminate form, one can take the derivative of the numerator and denominator separately to find a new limit: \( \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)} \). If this new limit can be evaluated or leads to another indeterminate form, L'Hôpital's Rule can be applied repeatedly until a determinate value is found or until it becomes clear that the limit does not exist.

Sine Function Behavior

The sine function is a periodic function with a range of [-1, 1]. As the values of x increase or decrease, the value of \( \sin(x) \) oscillates between these bounds. This is particularly important when evaluating limits involving trigonometric functions.

In our exercise, as \( x \rightarrow \infty \), \( \sin(x) \) does not approach a single value, but rather continues to oscillate. This can complicate the understanding of limits involving the sine function. However, when compared to functions with unbounded growth such as the exponential function, the bounded nature of the sine function often causes it to become insignificant in terms of contributing to the limit's final value, especially when it is in the numerator.

Exponential Function Growth

The exponential function, often represented as \( e^x \), is known for its rapid growth as x increases. Unlike the sine function which oscillates, \( e^x \) increases without any upper bound as \( x \rightarrow \infty \).

This characteristic leads to situations in evaluating limits where the exponential function's growth rate dominantly decides the behavior of a quotient. In the provided exercise, \( e^x \) in the denominator grows much faster than the oscillating \( \sin(x) \) in the numerator, causing the fraction as a whole to approach zero, illustrating the negligible impact of a bounded function compared to exponential growth.

Evaluating Limits

Evaluating limits is a fundamental aspect of calculus that deals with understanding the behavior of functions as they approach a specific point. This involves analyzing the tendency of the function's output as the input either gets arbitrarily large (as in \( x \rightarrow \infty \) ) or approaches a specific value.

To determine the value of a limit, analysts may employ various techniques ranging from direct substitution to more complex methods such as L'Hôpital's Rule. The goal is to determine the function's behavior near the point of interest and whether the function approaches a specific value, diverges to infinity, or oscillates without approaching any fixed value. The evaluation of the limit in our exercise successfully implemented L'Hôpital's Rule to show that as \( x \rightarrow \infty \) for the function \( \frac{\sin x}{e^x} \), the limit is 0, indicating that the exponential function's denominator grows too fast for the sine function numerator to influence the quotient's outcome.