The limit represents a derivative *𝑓*′(*𝑎*).f′(a). Find *𝑓*f and *𝑎*.and a.

lim*ℎ*→0sin(5*𝜋*3+*ℎ*)+3√2*ℎ*limh→0sin⁡(5π3+h)+32h

(Express numbers in exact form. Use symbolic notation and fractions where needed.)

*𝑓*(*𝑥*)=f(x)=

*𝑎*=a= make sure sqrt 2 is the denominator of 3

We are given a limit expression that represents the derivative \( f'(a) \). To find \( f(x) \) and \( a \), we can recognize that the limit represents the definition of the derivative:

\[
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
\]

The given limit expression is:

\[
\lim_{h \to 0} \frac{\sin\left(\frac{5\pi}{3} + h\right) + \frac{3}{\sqrt{2}}}{h}
\]

This suggests the form of the derivative of some function \( f(x) \), where \( f(x) = \sin(x) + \frac{3}{\sqrt{2}}x \), evaluated at \( a = \frac{5\pi}{3} \). Let's break this down:

### Step 1: Identify the function \( f(x) \)
In the expression, the term inside the sine function matches the argument \( \frac{5\pi}{3} + h \), which suggests that \( f(x) = \sin(x) \) is part of the function. The other term \( \frac{3}{\sqrt{2}} \) appears as a constant multiple of \( h \), implying a linear term in the function.

Thus, we can identify the function as:

\[
f(x) = \sin(x) + \frac{3}{\sqrt{2}} x
\]

### Step 2: Identify \( a \)
From the sine term \( \frac{5\pi}{3} + h \), we recognize that the point \( a \) at which the derivative is being evaluated is \( a = \frac{5\pi}{3} \).

### Final Answer
The function \( f(x) \) and point \( a \) are:

\[
f(x) = \sin(x) + \frac{3}{\sqrt{2}} x
\]
\[
a = \frac{5\pi}{3}
\]

---

Would you like further details on how to verify this or explore its derivative? 

Here are 5 related questions to consider:
1. What is the derivative of \( \sin(x) + \frac{3}{\sqrt{2}}x \)?
2. How does the chain rule apply when differentiating trigonometric functions?
3. Can you find the second derivative of \( f(x) \) at \( a \)?
4. How would the limit change if \( f(x) \) included a quadratic term?
5. What happens if we change the constant \( \frac{3}{\sqrt{2}} \)?

**Tip:** The derivative of a sine function \( \sin(x) \) is \( \cos(x) \), while the derivative of a linear term is its coefficient.

The limit represents a derivative f′(a). Find f and a. lim h→0 sin(5π/3 + h) + 3√2 h limh→0 sin⁡(5π/3 + h) + 3/√2h (Express numbers in exact form. Use symbolic notation and fractions where needed.) f(x)= a=

Learn how to find the function f(x) and the point a in a limit expression representing a derivative. This calculus problem involves limits, trigonometric functions, and linear terms, providing a detailed step-by-step solution. Suitable for advanced high school students in Grades 11-12.

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