If f(x) = 3x10 -7x + 5x-21x³ +3x²-7, then the value of limh->0 ((f(1-h)-f(1))/(h³+3h))

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.

Learn how to evaluate a limit involving the polynomial function f(x) = 3x^10 - 2x - 21x^3 + 3x^2 - 7. This problem involves using the definition of limits, polynomial evaluation, and binomial expansion approximation. Suitable for college-level mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation