Question

Prove that a function \(f\) is differentiable at \(x_{0}\) if and only if $$ \lim _{x \rightarrow x_{0}} \frac{f(x)-f\left(x_{0}\right)-m\left(x-x_{0}\right)}{x-x_{0}}=0 $$ for some constant \(m .\) In this case, \(f^{\prime}\left(x_{0}\right)=m\).

Short answer

Question: Prove that a function \(f\) is differentiable at a point \(x_0\) with derivative \(f^{\prime}(x_0) = m\) if and only if $$ \lim _{x \rightarrow x_{0}} \frac{f(x)-f(x_0)-m(x-x_0)}{x-x_0} = 0. $$ Answer: We have shown that if the given condition is true, then the function is differentiable at \(x_0\) with derivative \(f^{\prime}(x_0) = m\), and conversely, if the function is differentiable at \(x_0\), then the given condition is true. Therefore, the function is differentiable at \(x_0\) if and only if the given condition holds.

Step-by-step solution

Show the function is differentiable if the condition is true

Assume that $$ \lim _{x \rightarrow x_{0}} \frac{f(x)-f(x_0)-m(x-x_0)}{x-x_0} = 0. $$ We want to show that \(f\) is differentiable at \(x_0\). By definition, a function is differentiable at \(x_0\) if the following limit exists: $$ f^{\prime}(x_0) = \lim_{x\rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}. $$ We can rewrite this limit by adding and subtracting the term \(m(x-x_0)\) in the numerator: $$ f^{\prime}(x_0) = \lim_{x\rightarrow x_0} \frac{f(x)-f(x_0) - m(x-x_0) + m(x-x_0)}{x-x_0}. $$ Now we can split this limit into two separate limits: $$ f^{\prime}(x_0) = \lim_{x\rightarrow x_0} \frac{f(x)-f(x_0) - m(x-x_0)}{x-x_0} + \lim_{x\rightarrow x_0} \frac{m(x-x_0)}{x-x_0}. $$ Since we assumed that the first limit is 0 and the second one is equal to \(m\), we can conclude that: $$ f^{\prime}(x_0) = 0 + m = m. $$ This shows that the function is differentiable at \(x_0\) if the given condition is true.

Show the condition is true if the function is differentiable

Now let's assume that \(f\) is differentiable at \(x_0\) with derivative \(f^{\prime}(x_0) = m\). By the definition of differentiability, we have: $$ f^{\prime}(x_0) = \lim_{x\rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0} = m. $$ Similar to Step 1, we can add and subtract the term \(m(x-x_0)\) in the numerator: $$ \lim_{x\rightarrow x_0} \frac{f(x)-f(x_0) - m(x-x_0) + m(x-x_0)}{x-x_0} = m. $$ Now, we can split this limit into two separate limits: $$ \lim_{x\rightarrow x_0} \frac{f(x)-f(x_0) - m(x-x_0)}{x-x_0} + \lim_{x\rightarrow x_0} \frac{m(x-x_0)}{x-x_0} = m. $$ Since the second limit is equal to \(m\), we can subtract it from both sides of the equation: $$ \lim _{x \rightarrow x_{0}} \frac{f(x)-f(x_0)-m(x-x_0)}{x-x_0} = 0. $$ This shows that the given condition is true if the function is differentiable at \(x_0\).

Conclusion

We have shown that if the given condition is true, then the function is differentiable at \(x_0\) with derivative \(f^{\prime}(x_0) = m\), and conversely, if the function is differentiable at \(x_0\), then the given condition is true. Therefore, the function is differentiable at \(x_0\) if and only if the given condition holds.

Key concepts

Limits

In mathematical analysis, limits are fundamental to understanding how functions behave near specific points or as inputs grow large. When we talk about the limit of a function as it approaches a particular point, we describe how the function values converge to a particular number as the inputs get closer to that point. Limits deal with the behavior of functions elegantly and predict their tendencies with rigorous precision.

• A limit is essential in defining the derivative (hence differentiability) of a function at a given point.
• If \( \lim_{x \to x_0} f(x) = L \), it means the values of \( f(x) \) are getting closer to \( L \) as \( x \) approaches \( x_0 \).
• Limits can exist at finite values or as \( x \to \pm \infty \).

In our exercise, the condition \( \lim _{x \rightarrow x_{0}} \frac{f(x)-f(x_0)-m(x-x_0)}{x-x_0}=0 \) is crucial to proving differentiability because it captures the core idea of how the slope \( m \) shapes the function's linear behavior near \( x_0 \). Understanding limits is vital for grasping why this specific condition is linked to whether a function is differentiable.

Derivative

The derivative of a function at a particular point gives us the best linear approximation of that function near the point. In simple terms, it tells us how the function's output changes as the input changes, representing the slope of the function's graph at that point.

• The derivative, represented as \( f'(x_0) \) or \( \frac{df}{dx}(x_0) \), is defined as \( \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} \).
• A function is differentiable at a point if this limit exists.
• In the provided problem, \( f'(x_0) = m \) highlights that the condition given for differentiability essentially hinges on the definition of the derivative.

In essence, our exercise teaches that the derivative exists at \( x_0 \) only if the function behaves like a straight line with slope \( m \) when looked at closely near \( x_0 \). Hence, in differentiability, the concept of the derivative is both a tool and a target.

Real Analysis

Real analysis is a branch of mathematics focused on real-valued sequences, limits, and functions. It's the backbone that provides rigorous foundations for calculus. Key topics include not just limits and derivatives but also concepts like continuity and convergence.

• This field helps ensure that calculus assertions are deeply scrutinized and logically sound.
• It introduces the rigorous definitions for limits, continuity, and differentiability to make more abstract results tangible and applicable.
• Our exercise reflects the methods of real analysis, particularly in using sequence limits to study function properties like differentiability.

By understanding real analysis, the assumption that a function's differentiability implies a specific limit condition becomes more intuitive. Real analysis bridges the gap between abstract definitions and their practical implications in calculus and applied mathematics.