We are asked to find the value of the following limit:

$\lim_{h \to 0} \frac{f(1-h) - f(1)}{h^3 + 3h}$

where the function $f(x) = 3x^{10} - 7x + 5x - 21x^3 + 3x^2 - 7$.

Step 1: Simplify $f(x)$

First, let's simplify the expression for $f(x)$:

$f(x) = 3x^{10} - 7x + 5x - 21x^3 + 3x^2 - 7$ $f(x) = 3x^{10} - 2x - 21x^3 + 3x^2 - 7$

Step 2: Calculate $f(1-h)$ and $f(1)$

Next, let's evaluate $f(1)$ and $f(1-h)$:

$f(1) = 3(1)^{10} - 2(1) - 21(1)^3 + 3(1)^2 - 7$ $f(1) = 3 - 2 - 21 + 3 - 7 = -24$

Now, we need to expand $f(1-h)$. For small $h$, use the binomial expansion to approximate:

$f(1-h) = 3(1-h)^{10} - 2(1-h) - 21(1-h)^3 + 3(1-h)^2 - 7$

Expanding each term:

(1-h)^{10} \approx 1 - 10h + \frac{45h^2}{2} \quad \text{(we only consider terms up to h^2)} $(1-h)^3 \approx 1 - 3h + 3h^2$ $(1-h)^2 \approx 1 - 2h + h^2$

Substituting these into $f(1-h)$:

$f(1-h) \approx 3(1 - 10h + \frac{45h^2}{2}) - 2(1 - h) - 21(1 - 3h + 3h^2) + 3(1 - 2h + h^2) - 7$ $= 3 - 30h + \frac{135h^2}{2} - 2 + 2h - 21 + 63h - 63h^2 + 3 - 6h + 3h^2 - 7$ $= -24 + 29h + \left(\frac{135h^2}{2} - 60h^2\right)$ $= -24 + 29h + \frac{15h^2}{2}$

Step 3: Compute $f(1-h) - f(1)$

The difference $f(1-h) - f(1)$ is:

$f(1-h) - f(1) = \left(-24 + 29h + \frac{15h^2}{2}\right) - (-24)$ $= 29h + \frac{15h^2}{2}$

Step 4: Find the limit

Substitute into the limit expression:

$\lim_{h \to 0} \frac{29h + \frac{15h^2}{2}}{h^3 + 3h}$

Factor $h$ from both the numerator and denominator:

$\lim_{h \to 0} \frac{h \left(29 + \frac{15h}{2}\right)}{h(h^2 + 3)}$ $= \lim_{h \to 0} \frac{29 + \frac{15h}{2}}{h^2 + 3}$

As $h$ approaches 0:

$\lim_{h \to 0} \frac{29 + 0}{0 + 3} = \frac{29}{3}$

Thus, the value of the limit is:

$\boxed{\frac{29}{3}}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you use the binomial expansion to approximate functions?
2. What is the significance of using limits to find the slope of a tangent line?
3. How do higher-order terms affect the behavior of a function as $h$ approaches zero?
4. When can L'Hôpital's Rule be applied to evaluate limits?
5. What are some common techniques for simplifying complex functions before taking limits?

Tip: When evaluating limits involving polynomials, it’s often helpful to expand terms using binomial expansion, especially when $h$ is small.