Problem 39 - Technical Mathematics with Calculus

Question

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{(x+d)^{2}-2(x+d)-x^{2}+2 x}{d}$$

Step-by-Step Solution

Answer

\( \lim _{d \rightarrow 0} \frac{(x+d)^{2}-2(x+d)-x^{2}+2 x}{d} = 2x - 2 \)

Step 1: Expand the Numerator

First, expand the numerator by squaring the binomial and distributing the -2 across the term (x+d). This gives you: \( (x+d)^2 = x^2+2xd+d^2 \) and \( -2(x+d) = -2x-2d \) so the expanded numerator looks like \( x^2 + 2xd + d^2 - 2x - 2d \) .

Step 2: Combine Like Terms

Combine like terms within the expanded numerator. The terms \( x^2 \) and \( -x^2 \) cancel each other out, as do \( -2x \) and \( 2x \) . This simplifies the numerator to \( 2xd + d^2 - 2d \) .

Step 3: Factor Out the Common 'd'

Factor out 'd' from all terms in the numerator which results in: \( d(2x + d - 2) \) .

Step 4: Simplify the Fraction

Simplify the fraction by canceling the 'd' in the numerator with the 'd' in the denominator. This leaves \( 2x + d - 2 \) .

Step 5: Take the Limit

Take the limit of the simplified expression as \( d \) approaches 0, which leaves us with \( 2x + 0 - 2 \) , simplifying further to \( 2x - 2 \) .

Key Concepts

Limits of Expressions

Understanding the concept of a limit is central to calculus and helps us deal with situations where we want to find the value of a function as it approaches a certain point. When calculating the limit of an expression algebraically, we are interested in what happens to that expression as the variable, in this case, 'd', approaches a certain value - often 0 or infinity.

Using algebra, we manipulate the expression to avoid undefined or indeterminate forms such as 0/0 or infinity/infinity. In the exercise given, we are finding the limit as 'd' approaches 0 of a difference quotient, which is a common concept when dealing with derivatives - the rate at which a function changes at any given point.

Algebraic Simplification

The process of algebraic simplification involves combining like terms and reducing expressions to their simplest form. This is an important step before we can compute the limit of an expression. Simplification might involve expanding products and eliminating expressions that cancel each other out, like the terms involving x² and 2x in our example.

This step is critical for revealing the true nature of the function as the variable approaches the value of interest. In this case, combining like terms allowed us to clearly see which terms in the numerator would be affected as 'd' approaches zero, thereby revealing the essence of the expression's behavior at that point.

Factoring Algebraic Expressions

Another vital algebraic skill is factoring algebraic expressions. Factoring involves rewriting a sum or difference of terms as a product of factors. It can reveal common terms that might not be initially apparent and can be used to simplify complex algebraic expressions.

For instance, in our given problem, after combining like terms, we were left with a common 'd' that could be factored out. This is a key maneuver in limit problems because it often leads to the cancellation of terms that were causing unclear or undefined behavior. After factoring, the resulting expression was much easier to handle algebraically and allowed us to complete the limit evaluation cleanly by canceling out the common factor with the denominator.