a) Finding if lim𝑓(𝑥) as 𝑥 → 0 and lim𝑔(𝑥) as 𝑥 → 0 both do not exist:

Yes, it is possible for both limits to not exist. For example, consider two functions:

• $f(x) = \sin\left(\frac{1}{x}\right)$ for $x \neq 0$ and $f(0)$ is undefined. As $x \to 0$, the sine function oscillates between -1 and 1 infinitely, so the limit of $f(x)$ as $x \to 0$ does not exist.
• $g(x) = \frac{1}{x}$ for $x \neq 0$ and $g(0)$ is undefined. As $x \to 0$, the function approaches infinity, and thus the limit does not exist either.

Example functions where the limits do not exist:

• $f(x) = \sin\left(\frac{1}{x}\right)$
• $g(x) = \frac{1}{x}$

b) Using the chain rule to find the derivative of $\sqrt[3]{x^4}$:

To find the derivative of $y = \sqrt[3]{x^4}$, first rewrite it as:

$y = \left( x^4 \right)^{1/3} = x^{4/3}$

Now, apply the chain rule:

$\frac{dy}{dx} = \frac{d}{dx} \left( x^{4/3} \right)$

Using the power rule, we get:

$\frac{dy}{dx} = \frac{4}{3}x^{(4/3)-1} = \frac{4}{3}x^{1/3}$

So the derivative is:

$\frac{dy}{dx} = \frac{4}{3}x^{1/3}$

c) Finding the equation for the tangent line to the curve $y = f(x^2)$ at $(x, y) = (3, 7)$:

Given that $f(3) = 5$, $f(9) = 7$, $f'(3) = 5$, and $f'(9) = 13$, we need to find the tangent line to the curve $y = f(x^2)$ at the point $(3, 7)$.

We start by applying the chain rule to differentiate $y = f(x^2)$. Let $u = x^2$, so $y = f(u)$. Then, using the chain rule:

$\frac{dy}{dx} = f'(u) \cdot \frac{du}{dx}$

We know $u = x^2$, so:

$\frac{du}{dx} = 2x$

Thus, the derivative of $y$ is:

$\frac{dy}{dx} = f'(x^2) \cdot 2x$

Now, evaluate this derivative at $x = 3$:

$\frac{dy}{dx}\Big|_{x=3} = f'(9) \cdot 2(3) = 13 \cdot 6 = 78$

Now we know the slope of the tangent line at $x = 3$ is 78. The point of tangency is $(3, 7)$.

So, using the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

Substitute $m = 78$, $x_1 = 3$, and $y_1 = 7$:

$y - 7 = 78(x - 3)$

Simplifying:

$y - 7 = 78x - 234$

$y = 78x - 227$

So, the equation of the tangent line is:

$y = 78x - 227$

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Would you like further details on any of these solutions? Or do you have any other questions?

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Here are some additional questions to explore:

1. How do limits behave for oscillatory functions like $\sin\left( \frac{1}{x} \right)$?
2. What are other examples of functions that do not have limits as $x \to 0$?
3. Can you apply the chain rule to more complex compositions, such as $f(g(x))$ where both functions are non-polynomial?
4. How can you use the power rule for derivatives with fractional exponents?
5. What is the geometric interpretation of the derivative of a function at a point?

Tip: When working with the chain rule, remember to treat composite functions step-by-step, differentiating the outer function first and then multiplying by the derivative of the inner function.