https://file.aiquickdraw.com/mathbot/file/1729243947207f6lla668.jpg

The problem asks us to determine the value of \(\phi\) such that the given function \(f(x)\) is continuous at \(x = 0\). The function is piecewise, and it is defined as:

\[
f(x) = 
\begin{cases} 
\frac{2x + \sin^{-1}(x)}{2x - \tan^{-1}(x)}, & x \neq 0 \\
2\phi - 1, & x = 0
\end{cases}
\]

To ensure the function is continuous at \(x = 0\), we need to make sure that:

\[
\lim_{x \to 0} f(x) = f(0)
\]

### Step 1: Find \(\lim_{x \to 0} f(x)\) for \(x \neq 0\)

We will compute the limit of the expression for \(f(x)\) as \(x \to 0\):

\[
\lim_{x \to 0} \frac{2x + \sin^{-1}(x)}{2x - \tan^{-1}(x)}
\]

For small values of \(x\), the approximations \(\sin^{-1}(x) \approx x\) and \(\tan^{-1}(x) \approx x\) hold. Applying these approximations:

\[
\lim_{x \to 0} \frac{2x + x}{2x - x} = \lim_{x \to 0} \frac{3x}{x} = 3
\]

Thus:

\[
\lim_{x \to 0} f(x) = 3
\]

### Step 2: Continuity condition at \(x = 0\)

For \(f(x)\) to be continuous at \(x = 0\), we must have:

\[
f(0) = \lim_{x \to 0} f(x)
\]

Given that \(f(0) = 2\phi - 1\), we set:

\[
2\phi - 1 = 3
\]

### Step 3: Solve for \(\phi\)

Now, solve for \(\phi\):

\[
2\phi = 4 \quad \Rightarrow \quad \phi = 2
\]

Thus, the value of \(\phi\) that ensures continuity at \(x = 0\) is \(\boxed{2}\).

---

Do you want further clarification or details on any part of this solution? Here are some related questions to explore:

1. What are some other common methods for ensuring the continuity of piecewise functions?
2. Why do we use small approximations like \(\sin^{-1}(x) \approx x\) as \(x \to 0\)?
3. What would happen if we chose a different value of \(\phi\)?
4. How do limits help in defining the continuity of a function?
5. Can a function be continuous but not differentiable? Can you provide an example?

**Tip:** Always check for both one-sided limits when determining the continuity of a piecewise function at a specific point.

Determine the value of \(\phi\) given that the following function is continuous at \(x = 0\):
\[ f(x) = \begin{cases} \frac{2x + \sin^{-1}(x)}{2x - \tan^{-1}(x)}, & x \neq 0 \\ 2\phi - 1, & x = 0 \end{cases} \]

This problem explores how to find the value of \(\phi\) that ensures the continuity of a piecewise function at \(x = 0\). Key concepts include limits, trigonometric approximations, and the continuity theorem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation