Question

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{2}-2(x+\Delta x)+1-\left(x^{2}-2 x+1\right)}{\Delta x} $$

Short answer

The limit is 2.

Step-by-step solution

Simplify the Expression inside the Limit Sign

First, simplify the expression inside the limit sign. We can simplify the terms in the top part of the fraction by arranging the like terms and performing the subtraction. Simplify \( (x+\Delta x)^{2}-2(x+\Delta x)+1 \) and \( x^{2}-2x+1 \). The resulting expression will be \( \lim _{\Delta x \rightarrow 0} \frac{2 \Delta x+\Delta x^{2}}{\Delta x} \).

Simplify the Fraction

Next, we can further simplify the fraction by dividing each term in the numerator by \( \Delta x \). This results in \( \lim _{\Delta x \rightarrow 0} \frac{2+\Delta x}{1} \).

Apply the Limit

Apply the limit to the simplified expression. As \( \Delta x \) approaches 0, \( \Delta x \) itself tends towards 0. This results in \( \lim _{\Delta x \rightarrow 0} 2+\Delta x = 2+0 = 2 \).

Key concepts

Derivatives

Derivatives are a fundamental concept in calculus that describe how a function changes as its input changes. Simply put, the derivative of a function provides the rate at which one variable changes with respect to another. For instance, if you have a function that shows time versus position, the derivative will tell you the object's velocity.

To find a derivative, you typically use the limit process. This involves examining the behavior of the function as inputs get closer to a specific point. In our exercise, we are using the difference quotient, which is a classic way to introduce derivatives.

• The difference quotient: \ \[ \frac{f(x+\Delta x) - f(x)}{\Delta x} \]
• As \ \Delta x \ becomes infinitely small, this expression approximates the derivative of the function at a point \ x \.

Taking derivatives is an essential tool for understanding motion, rates of change, optimization, and other real-life applications. Mastering derivatives opens up new ways to see and understand the world through the lens of calculus.

Calculus

Calculus is the branch of mathematics that deals with continuous change. It is divided mainly into two parts: differentiation and integration. In this problem, we focus on differentiation, which is all about finding derivatives.

Differentiation can be thought of as a formal way of finding the rate of change or slope of a graph at any given point. This is crucial in many fields such as physics, engineering, and economics for modeling and solving real-world problems.

The foundational idea in calculus used here is the limit. The limit allows us to "zoom in" on a function at a certain point to understand its behavior there without directly substituting extreme values, like infinity or zero.

• Key concepts include:
  • Limits: Approaching a specific value, rather than hitting it exactly.
  • Derivatives: The slope of the tangent line to the curve at a point.

By fully grasping these concepts, you can solve complex equations and understand dynamic systems. Calculus acts as a bridge between algebra and the real world, giving you powerful tools to solve problems.

Algebraic Simplification

Algebraic simplification is a necessary skill in solving calculus problems, especially when finding limits and derivatives. Simplification involves rewriting an expression in a simpler or more convenient form without changing its value.

In our exercise, we first simplify the expression inside the limit before even applying calculus principles. This begins with simplifying the polynomial formed after expansion, which results in the neat separation of terms.

• Steps in algebraic simplification:
  • Expand expressions: Multiply out any expressions within brackets.
  • Combine like terms: Group similar terms together to simplify the expression.
  • Cancel common factors: Reduce fractions by canceling terms that appear in both the numerator and the denominator.

Through algebraic simplification, we make calculus problems more manageable and easier to solve. It clears unnecessary complications and helps focus on the core principles of the problem.