16. If f is differentiable at xo, prove that lim h→0 f(xo + αh) - f(xo - ßh) h = (x + ß)ƒ'(xo).
16. If f is differentiable at xo, prove that lim h→0 f(xo + αh) - f(xo - ßh) h = (x + ß)ƒ'(xo).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem 16**
If \( f \) is differentiable at \( x_0 \), prove that
\[
\lim_{h \to 0} \frac{f(x_0 + \alpha h) - f(x_0 - \beta h)}{h} = (\alpha + \beta) f'(x_0).
\]
---
**Explanation:**
This problem involves proving a limit related to the differentiability of a function at a point. Given that \( f \) is differentiable at \( x_0 \), the challenge is to show that the limit expression equates to \( (\alpha + \beta) f'(x_0) \).
In this context:
- \( f'(x_0) \) denotes the derivative of \( f \) at \( x_0 \).
- Values \( \alpha \) and \( \beta \) are constants that scale the small increment \( h \).
This type of limit is a generalization of the standard definition of the derivative, which involves taking a limit as \( h \) approaches zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cfc5ac-076f-4cb7-b69f-6b7f1cfee42f%2F5645476d-24eb-4472-b802-28a5129ecb9f%2F5dtyjp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 16**
If \( f \) is differentiable at \( x_0 \), prove that
\[
\lim_{h \to 0} \frac{f(x_0 + \alpha h) - f(x_0 - \beta h)}{h} = (\alpha + \beta) f'(x_0).
\]
---
**Explanation:**
This problem involves proving a limit related to the differentiability of a function at a point. Given that \( f \) is differentiable at \( x_0 \), the challenge is to show that the limit expression equates to \( (\alpha + \beta) f'(x_0) \).
In this context:
- \( f'(x_0) \) denotes the derivative of \( f \) at \( x_0 \).
- Values \( \alpha \) and \( \beta \) are constants that scale the small increment \( h \).
This type of limit is a generalization of the standard definition of the derivative, which involves taking a limit as \( h \) approaches zero.
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