Let f(x) = 3x^{10} - 7x^{8} + 5x^{6} - 21x^{3 | vedclass

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(C) Given $f(x) = 3x^{10} - 7x^{8} + 5x^{6} - 21x^{3} + 3x^{2} - 7$. We need to evaluate the limit $L = \lim_{h \rightarrow 0} \frac{f(1-h) - f(1)}{h^

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is $\frac{53}{3}$

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(C) Given $f(x) = 3x^{10} - 7x^{8} + 5x^{6} - 21x^{3} + 3x^{2} - 7$. We need to evaluate the limit $L = \lim_{h \rightarrow 0} \frac{f(1-h) - f(1)}{h^{3} + 3h}$. We can rewrite the expression as: $L = \lim_{h \rightarrow 0} \left( \frac{f(1-h) - f(1)}{-h} \cdot \frac{-h}{h(h^{2} + 3)} \right)$. Since $f'(1) = \lim_{h \rightarrow 0} \frac{f(1-h) - f(1)}{-h}$, we have: $L = f'(1) \cdot \lim_{h \rightarrow 0} \frac{-1}{h^{2} + 3}$. First, calculate the derivative $f'(x) = 30x^{9} - 56x^{7} + 30x^{5} - 63x^{2} + 6x$. Evaluating at $x = 1$: $f'(1) = 30(1)^{9} - 56(1)^{7} + 30(1)^{5} - 63(1)^{2} + 6(1) = 30 - 56 + 30 - 63 + 6 = -53$. Now substitute this into the limit expression: $L = (-53) \cdot \left( \frac{-1}{0^{2} + 3} \right) = (-53) \cdot \left( -\frac{1}{3} \right) = \frac{53}{3}$.

Let $f(x) = 3x^{10} - 7x^{8} + 5x^{6} - 21x^{3} + 3x^{2} - 7$. Then $\lim_{h \rightarrow 0} \frac{f(1-h) - f(1)}{h^{3} + 3h}$ is:

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