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 limit:

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

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

### Step 1: Simplify \( f(x) \)
First, simplify the function \( f(x) \):

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

### Step 2: Calculate \( f(1-h) \) and \( f(1) \)
Next, substitute \( 1-h \) into the function to find \( f(1-h) \):

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

And for \( f(1) \):

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

### Step 3: Expand \( f(1-h) \) for small \( h \)
Now, expand \( f(1-h) \) using the binomial theorem:

\[
(1-h)^{10} = 1 - 10h + \frac{45h^2}{2} - \dots
\]
\[
(1-h)^3 = 1 - 3h + 3h^2 - h^3
\]
\[
(1-h)^2 = 1 - 2h + h^2
\]

Substituting these expansions into \( f(1-h) \):

\[
f(1-h

Explore how to calculate the limit lim h -> 0 ((f(1-h) - f(1)) / (h^3 + 3h)) using the function f(x) = 3x^10 - 2x - 21x^3 + 3x^2 - 7. This problem involves limits, differentiation, and the binomial theorem, suitable for undergraduate-level mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation