Problem 15 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{\Delta x \rightarrow 0^{-}} \frac{\frac{1}{x+\Delta x}-\frac{1}{x}}{\Delta x} $$

Step-by-Step Solution

Answer

The limit of the given expression as \( \Delta x \rightarrow 0^{-} \) is \( \frac{-1}{x^2} \).

Step 1: Simplify the numerator

Begin by simplifying the numerator of the given fractional expression. Since it is in the form of a fraction of fractions, this can be simplified by finding a common denominator and combining the two fractions in the numerator which results to \( \frac{x-(x+\Delta x)}{x(x + \Delta x)} \), which further simplifies to \( \frac{-\Delta x}{x(x + \Delta x)} \). Thus, the full expression becomes \( \lim _{\Delta x \rightarrow 0^{-}} \frac{\frac{-\Delta x}{x(x + \Delta x)}}{\Delta x} \).

Step 2: Simplify the entire expression

Next step is to simplify the entire expression. The fractional expression can be simplified by canceling out the \(\Delta x\) terms from the numerator and the denominator of the entire fraction, leaving \( \lim _{\Delta x \rightarrow 0^{-}} \frac{-1}{x(x + \Delta x)} \).

Step 3: Compute the limit

With the expression simplified, the limit can now be computed. Substitute \( \Delta x = 0 \) into the expression, yielding \( \frac{-1}{x^2} \) as the final result.

Key Concepts

Numerator Simplification

When dealing with limits involving fractional expressions, simplifying the numerator often becomes the first critical step. In our exercise, the original numerator consists of a difference of two fractions, expressed as \( \frac{1}{x+\Delta x} - \frac{1}{x} \). To simplify this, we need to consolidate these fractions into a single expression.

To achieve this, we identify a common denominator, which in this case is \( x(x + \Delta x) \). By rewriting each fraction with this common denominator, we can combine them effectively:

• Rewrite \( \frac{1}{x+\Delta x} \) as \( \frac{x}{x(x + \Delta x)} \)
• Rewrite \( \frac{1}{x} \) as \( \frac{(x + \Delta x)}{x(x + \Delta x)} \)

Now, the expression becomes \( \frac{x - (x + \Delta x)}{x(x + \Delta x)} \). Simplifying further, it leads to \( \frac{-\Delta x}{x(x + \Delta x)} \), which is a simpler form making it easier to work with in later steps.

Fractional Expression

A fractional expression can appear complicated, particularly when dealing with limits. These expressions often include fractions both in the numerator and the denominator.

In our example, once we've simplified the numerator, the expression turns into \( \lim _{\Delta x \rightarrow 0^{-}} \frac{\frac{-\Delta x}{x(x + \Delta x)}}{\Delta x} \). It’s crucial to understand how to handle these types of expressions rigorously.

You can usually simplify fractional expressions by strategically canceling out similar terms. For the given exercise, by noticing that \( \Delta x \) appears both in the numerator and the denominator, you can simplify the fraction by canceling these \( \Delta x \) terms. This reduces our expression to \( \lim _{\Delta x \rightarrow 0^{-}} \frac{-1}{x(x + \Delta x)} \). This step is fundamental as it directly connects to computing the limit later on.

Common Denominator

Using a common denominator is an essential technique in simplifying expressions with fractions in the numerator or denominator. When fractions need to be added or subtracted, you convert them into fractions with the same denominator, which allows for direct arithmetic operations.

For example, consider \( \frac{1}{x+\Delta x} \) and \( \frac{1}{x} \). Their common denominator is \( x(x + \Delta x) \).

With this common denominator, the fractions can be rewritten and combined:

• \( \frac{1}{x+\Delta x} = \frac{x}{x(x+\Delta x)} \)
• \( \frac{1}{x} = \frac{x + \Delta x}{x(x + \Delta x)} \)

This simplification through a common denominator turns a challenging problem into one that is more manageable and easier to solve, laying a foundation for computing the limit.

Computing Limits

After a fractional expression is fully simplified, computing the limit is often the final step. This is where the initial problem is solved through limiting behavior as a variable approaches a certain value.

In the exercise, once we’ve simplified the expression to \( \lim _{\Delta x \rightarrow 0^{-}} \frac{-1}{x(x + \Delta x)} \), computing the limit involves directly substituting \( \Delta x = 0 \) into the function.

Here are the steps involved:

• Substitute \( \Delta x = 0 \) into the expression: \( \frac{-1}{x(x + 0)} = \frac{-1}{x^2} \).
• The limit, therefore, results in \( \frac{-1}{x^2} \).

While this might seem straightforward, computing limits is a crucial part of calculus, serving as a foundation for defining derivatives and understanding function behavior at points.