Problem 32 - Calculus: Early Transcendentals

Question

Evaluate the limit, if it exists. \( \displaystyle \lim_{h \to 0}\frac{\frac{1}{(x + h)^2} - \frac{1}{x^2}}{h} \)

Step-by-Step Solution

Answer

The limit is \( \frac{-2}{x^3} \).

Step 1: Recognize the Limit as a Derivative

Notice that the expression resembles the definition of a derivative. Specifically, this limit is the derivative of the function \( f(x) = \frac{1}{x^2} \) at a point \( x \). The general derivative is given by:\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \] Thus, the expression given, \( \lim_{h \to 0} \frac{\frac{1}{(x + h)^2} - \frac{1}{x^2}}{h} \), represents \( f'(x) \).

Step 2: Find a Common Denominator

Rewriting the expression as a single fraction can simplify it. Here we have:\[ \frac{1}{(x+h)^2} - \frac{1}{x^2} = \frac{x^2 - (x+h)^2}{x^2 (x+h)^2} \]Expanding \((x + h)^2\) gives:\[ x^2 + 2xh + h^2 \]So our expression becomes:\[ \frac{x^2 - x^2 - 2xh - h^2}{x^2 (x+h)^2} = \frac{-2xh - h^2}{x^2 (x+h)^2} \]

Step 3: Factor out \( h \) from the Numerator

In the expression \( -2xh - h^2 \), factor out an \( h \):\[ -2xh - h^2 = h(-2x - h) \]This gives us:\[ \frac{h(-2x - h)}{x^2 (x + h)^2} \]

Step 4: Simplify and Cancel \( h \)

Now, simplify the expression by canceling the \( h \) in the numerator and the \( h \) in the denominator:\[ \frac{-2x - h}{x^2 (x + h)^2} \]

Step 5: Evaluate the Limit as \( h \to 0 \)

Once simplified, evaluate the limit as \( h \to 0 \):\[ \lim_{h \to 0} \frac{-2x - h}{x^2 (x + h)^2} = \frac{-2x}{x^2 \cdot x^2} = \frac{-2x}{x^4} = \frac{-2}{x^3} \]

Key Concepts

Derivative

The concept of a derivative is foundational in calculus. It represents the rate at which a function changes at any given point. In simpler terms, the derivative tells us "how steep" a function is at a specific point.
To find the derivative, we often use the definition involving limits and the difference quotient:

• The difference quotient is given by \( \frac{f(x+h) - f(x)}{h} \).
• To find the derivative \( f'(x) \), we evaluate the limit \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

To understand derivatives, consider a curve on a graph. Derivatives help us understand how this curve behaves at each point, such as determining whether a curve is increasing, decreasing, or staying flat.

Difference Quotient

The difference quotient is a crucial part of finding the derivative. It is an expression that represents the average rate of change of the function over a small interval.

• Its formula is \( \frac{f(x+h) - f(x)}{h} \).
• This measures the slope of the secant line connecting two points on the function, \((x, f(x))\) and \((x+h, f(x+h))\).

As \( h \) becomes very small and approaches zero, the secant line becomes closer to being a tangent line. Ultimately, the limit as \( h \to 0 \) gives us the slope of this tangent line, which is the derivative.

Rational Functions

Rational functions are quotients of polynomial functions. A typical form is \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. They can often be identified by the presence of variables in the denominator.

• In our example, \( f(x) = \frac{1}{x^2} \) is a rational function.
• The techniques for finding limits or derivatives for these functions involve algebraic manipulation, such as finding common denominators or factoring.

When working with rational functions, it's essential to ensure the denominator is not zero since division by zero in mathematics is undefined.

h Approaching Zero

Understanding \( h \) approaching zero is key in limit evaluation. In calculus, we use \( h \) as a small increment to examine how functions behave at a point when the increment becomes infinitesimally small.

• We often use the expression "as \( h \to 0 \)" to describe this process.
• This process helps transition from a discrete to a continuous perspective.

By evaluating the behavior of a function as \( h \) approaches zero, we gain insights into the instantaneous rate of change and the slope of the tangent line at any point on the function's graph. This is a fundamental aspect of understanding derivatives and their applications.