Problem 41 - Calculus Volume 1

Question

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=x^{1 / 3}, x=0 $$

Step-by-Step Solution

Answer

The derivative does not exist at \(x = 0\) because the limit tends to infinity.

Step 1 - Understand the Limit Definition of Derivative

The derivative of a function \(f(x)\) at point \(x = a\) is defined as the limit \( \lim_{{h \to 0}} \frac{{f(a+h) - f(a)}}{h} \), provided this limit exists. For this problem, we are checking if the derivative exists at \(x = 0\) for \(f(x) = x^{1/3}\).

Step 2 - Substitute the Function into the Definition

Substitute \(f(x) = x^{1/3}\) and \(a = 0\) into the limit definition of the derivative: \[ \lim_{{h \to 0}} \frac{{(0+h)^{1/3} - 0^{1/3}}}{h} = \lim_{{h \to 0}} \frac{{h^{1/3}}}{h}. \]

Step 3 - Simplify the Limit Expression

Simplify the expression: \[ \lim_{{h \to 0}} \frac{{h^{1/3}}}{h} = \lim_{{h \to 0}} h^{-2/3}. \]This is because \(h^{1/3} = h^{1/3} \times h^{-1} = h^{1/3 - 1} = h^{-2/3}\).

Step 4 - Evaluate the Limit

Evaluate the limit: \( \lim_{{h \to 0}} h^{-2/3} \). As \(h\) approaches 0, \(h^{-2/3}\) tends to infinity. Thus, the limit does not exist because an infinite limit indicates that the function is not differentiable at that point.

Key Concepts

Derivative Does Not Exist

In mathematics, the concept of a derivative is central to understanding how functions change at a specific point. However, there are situations where the derivative of a function does not exist at a certain point. This can occur when the function is not smooth or has abrupt changes in behavior at that point.

Consider the example provided: the function is given by \( f(x) = x^{1/3} \), and we are examining its derivative at \( x=0 \). To determine whether the derivative exists, we use the limit definition: \( \ \lim_{{h o 0}} \frac{{f(a+h) - f(a)}}{h} \).

When substituting into the definition, we end up with \( \ \lim_{{h \to 0}} \frac{{h^{1/3}}}{h} = \ \lim_{{h \to 0}} h^{-2/3} \). As \( h \) approaches zero, \( h^{-2/3} \) tends to infinity, which tells us that the limit does not exist. Consequently, the derivative at \( x=0 \) does not exist for this function.

The main takeaway is that a function may not have a derivative at certain points if the behavior of the function becomes non-linear or behaves discontinuously, as seen here.

Function Differentiability

For a function to be differentiable at a point, it means having a well-defined slope or rate of change at that specific point. Differentiability is closely tied to the existence of a derivative.

In our example of \( f(x) = x^{1/3} \), we're interested in knowing if it's differentiable at \( x=0 \). Using the limit definition confirms that the limit does not exist because it goes to infinity. This implies the function is not differentiable at this point.

Differentiability has a few key requirements:

• The function must be continuous at the point in question.
• There must be no sharp corners or cusps at the point.
• The left-hand derivative and right-hand derivative should be equal.

The cubic root function, \( x^{1/3} \), is continuous at \( x=0 \), but the slope approaches infinity, violating the smoothness needed for differentiability. Thus, smoothness or linearity of change is essential for a function to have a derivative at a given point.

Evaluating Limits

Evaluating limits is a fundamental aspect of calculus and is particularly crucial when using the limit definition of the derivative. In mathematical terms, evaluating a limit explores what happens to a function as the variable approaches a certain value.

For the derivative to exist, the limit \( \ \lim_{{h \to 0}} \frac{{f(a+h) - f(a)}}{h} \) must yield a finite number. If the expression approaches a specific number, then this number is the derivative.

In the case of \( f(x) = x^{1/3} \), the limit \( \ \lim_{{h \to 0}} h^{-2/3} \) does not reach a finite value; instead, it goes to infinity. This means the expression becomes unbounded.

Successfully evaluating the limit would have shown that the function's behavior at a point in terms of how abruptly it changes is manageable and linear. However, when the limit results in infinity or becomes undefined, it indicates non-differentiability at that point.