The goal is to show that the equation $e^x = \sin(x)$ has infinitely many solutions using the Intermediate Value Theorem (IVT). Let's break it down step by step.

Step 1: Understanding the Functions

1. The function $e^x$ is continuous and strictly increasing for all real $x$. As $x \to -\infty$, $e^x \to 0$, and as $x \to \infty$, $e^x \to \infty$.
2. The function $\sin(x)$ is continuous and oscillates between -1 and 1 for all real $x$, repeating its values periodically.

Step 2: Intermediate Value Theorem (IVT)

The IVT states that if a continuous function $f(x)$ takes values $f(a)$ and $f(b)$ at two points $a$ and $b$, and if a value $N$ lies between $f(a)$ and $f(b)$, then there exists at least one $c$ in the interval $(a, b)$ such that $f(c) = N$.

Step 3: Application of IVT

1. Define the function $f(x) = e^x - \sin(x)$. This function is continuous because both $e^x$ and $\sin(x)$ are continuous.
2. We need to show that $f(x) = 0$ (which corresponds to $e^x = \sin(x)$) has infinitely many solutions.

Step 4: Choosing Intervals to Apply IVT

• Since $e^x \to \infty$ as $x \to \infty$ and $\sin(x)$ is bounded between -1 and 1, the function $f(x) = e^x - \sin(x)$ grows without bound as $x \to \infty$.
• At $x \to -\infty$, $e^x \to 0$, while $\sin(x)$ still oscillates between -1 and 1, so $f(x) = e^x - \sin(x)$ will oscillate between values close to $-1$ and values close to $1$.

For large negative $x$, $e^x$ approaches zero, so $f(x) \approx -\sin(x)$. In particular:

• At some points, $\sin(x) = 1$, giving $f(x) \approx -1$.
• At some points, $\sin(x) = -1$, giving $f(x) \approx 1$.

Thus, for large negative values of $x$, $f(x)$ takes values on both sides of zero.

Step 5: Conclusion from IVT

Since $f(x)$ oscillates and takes both positive and negative values as $x \to -\infty$, and $f(x) \to \infty$ as $x \to \infty$, the IVT guarantees that there are infinitely many points where $f(x) = 0$, meaning $e^x = \sin(x)$ has infinitely many solutions.

This shows that the equation $e^x = \sin(x)$ has infinitely many solutions, without the need for a graph.

Do you have any questions or would you like further clarification?

Here are 5 related questions:

1. How does the behavior of the sine function differ from that of $e^x$ as $x$ increases or decreases?
2. Can we apply the IVT to any continuous function, or are there conditions that must be met?
3. How do periodic functions influence the application of the IVT?
4. Could a similar technique be used to find solutions to $e^x = \cos(x)$?
5. How would you find approximate values for some of the solutions of $e^x = \sin(x)$?

Tip: When using the Intermediate Value Theorem, ensure that the function is continuous over the interval you're examining.