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To solve this problem, we’ll go through it step-by-step.

### Given Information
The function \( f(x) = \sqrt[3]{x} \) and the derivative formula for \( f'(a) \):
\[
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
\]

### Step 1: Find \( f'(a) \) for \( a \neq 0 \)
For \( a \neq 0 \), we can calculate \( f'(a) \) by using the limit definition of the derivative.

1. Substitute \( f(x) = \sqrt[3]{x} \) into the formula for \( f'(a) \):
   \[
   f(a + h) = \sqrt[3]{a + h} \quad \text{and} \quad f(a) = \sqrt[3]{a}
   \]
   
   So, we get:
   \[
   f'(a) = \lim_{h \to 0} \frac{\sqrt[3]{a + h} - \sqrt[3]{a}}{h}
   \]

2. To evaluate this limit, let’s use the fact that the derivative of \( \sqrt[3]{x} \) with respect to \( x \) is \( \frac{1}{3}x^{-\frac{2}{3}} \). Using this directly:
   \[
   f'(x) = \frac{1}{3}x^{-\frac{2}{3}}
   \]
   
   So, for \( f'(a) \) when \( a \neq 0 \):
   \[
   f'(a) = \frac{1}{3}a^{-\frac{2}{3}}
   \]

### Step 2: Show that \( f'(0) \) does not exist and that \( f \) has a vertical tangent line at \( (0,0) \)
To investigate the existence of \( f'(0) \), we use the definition of the derivative at \( x = 0 \):
\[
f'(0) = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0} \frac{\sqrt[3]{h} - 0}{h} = \lim_{h \to 0} \frac{\sqrt[3]{h}}{h}
\]

Now, rewrite \( \sqrt[3]{h} = h^{\frac{1}{3}} \):
\[
f'(0) = \lim_{h \to 0} \frac{h^{\frac{1}{3}}}{h} = \lim_{h \to 0} h^{\frac{1}{3} - 1} = \lim_{h \to 0} h^{-\frac{2}{3}}
\]

Since \( h^{-\frac{2}{3}} \) approaches infinity as \( h \to 0 \), the limit does not exist. This implies that \( f'(0) \) does not exist.

### Conclusion: Vertical Tangent at \( (0,0) \)
The fact that \( f'(0) \) does not exist and the derivative approaches infinity as \( h \to 0 \) from either direction indicates that the graph of \( f(x) = \sqrt[3]{x} \) has a vertical tangent line at \( (0, 0) \).

Would you like further details on any of these steps?

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### Related Questions

1. How does one interpret a vertical tangent geometrically on a graph?
2. How can we find the derivative of other roots, such as \( \sqrt[4]{x} \)?
3. What does it mean for a derivative to be infinite at a certain point?
4. How would we apply this limit method for other non-integer power functions?
5. What would the tangent line look like if \( f(x) = x^{2/3} \)?

### Tip
When dealing with non-integer powers, converting terms to fractional exponents often simplifies the differentiation process.

Consider the following equation: Let f(x) = √(3)x. If a ≠ 0, use the above formula to find f'(a). Show that f'(0) does not exist and that f has a vertical tangent line at (0,0).

Learn how to find the derivative of f(x) = √(3)x for non-zero values and investigate the behavior at x = 0. This solution explores the use of the limit definition of the derivative, applying fractional exponents, and shows that f'(0) does not exist, resulting in a vertical tangent at (0,0). Suitable for college-level calculus students.

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