Problem 85 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(83-88\) , find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\) $$ f(x)=x^{2}-4 x $$

Step-by-Step Solution

Answer

The limit of the function as \(\Delta x\) approaches 0 is \(2x-4\).

Step 1: Substitute the function into the limit definition

Firstly, substitute for \(f(x + \Delta x)\) and \(f(x)\) in the limit formula. That would translate into \(\lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{2}-4*(x+\Delta x)-(x^{2}-4x)}{\Delta x}\). Then simplify to get \(\lim _{\Delta x \rightarrow 0} \frac{x^{2}+2*x*\Delta x+(\Delta x)^{2}-4*x-4*\Delta x-x^{2}+4x)}{\Delta x}\)

Step 2: Simplify the expression

After substituting, simplify the expression inside the limit operation. This leads to \(\lim _{\Delta x \rightarrow 0} \frac{2*x*\Delta x+(\Delta x)^{2}-4*\Delta x)}{\Delta x}\). We can then factor out \(\Delta x\), simplifying the expression further, yielding \(\lim _{\Delta x \rightarrow 0} 2*x+\Delta x-4\).

Step 3: Apply the Limit

Finally, allow \(\Delta x\) to approach 0 to compute the final limit. This gives \(2*x+0-4\), which simplifies to \(2x-4\).

Key Concepts

Limit Definition

Understanding the definition of a limit in calculus is fundamental for solving problems related to change and motion. A limit evaluates the value that a function approaches as the input (or the variable) gets infinitely close to a certain number. In mathematical terms, the notation \( \lim_{x \rightarrow c}f(x) \) expresses the limit of the function \( f(x) \) as \( x \) approaches \( c \). The limit tells us the value that \( f(x) \) gets closer to as \( x \) gets closer to \( c \), without necessarily reaching it. For example, if you have a function that represents the speed of a car, the limit can help you find the speed as you come infinitesimally close to a certain point in time. This concept is a cornerstone in calculus, as it allows for the understanding of derivatives and the behavior of functions at specific points or intervals.

Simplifying Expressions

Simplifying expressions is a process that rewrites complex mathematical phrases into a simpler or more manageable form. This often involves combining like terms, distributing products over sums, and canceling common factors. The goal is to make the expression less cumbersome and easier to work with, particularly in the context of limit operations. In the given example, simplification is used to minimize the terms in the numerator. This makes it easier to identify how \( \Delta x \) affects the limit and to factor \( \Delta x \) out of the expression before proceeding with the limit operation. Simplification doesn't change the fundamental nature or value of the expression; it just provides a clearer and more direct path to finding the limit.

Limit Operation

Performing a limit operation involves finding the value that a function approaches as the variable progresses towards a particular point. When executing a limit operation, you typically follow a series of steps: substitution, simplification, and in some cases, the application of special limit laws or properties. In cases where direct substitution results in an indeterminate form such as \( 0/0 \), additional algebraic manipulations, like factoring, may be required to resolve the limit. In this exercise, once the expression is simplified, the limit operation becomes straightforward: we allow \( \Delta x \) to approach zero. This step is pivotal in determining the behavior of \( f(x) \) with respect to \( x \) at that particular point and directly leads to finding the derivative of the function.

Factoring in Algebra

Factoring is a crucial algebraic process where we express an equation or expression as the product of its factors. These factors are usually simpler expressions or numbers which, when multiplied together, give the original expression. In the context of finding limits, factoring helps eliminate common factors between the numerator and denominator to resolve indeterminate forms or simplify expressions before applying limit operations. For the exercise at hand, factoring \( \Delta x \) out from the terms in the numerator simplifies the expression, making it possible to cancel it against the \( \Delta x \) in the denominator. This leaves us with a simpler function that is easier to handle when applying the limit operation as \( \Delta x \) approaches zero. Factoring can often be a gateway to revealing the structure of a function around a specific point, making it indispensable for solving limit problems.