Chapter 2: Problem 2
Evaluate \(\lim _{x \rightarrow 1}\left(x^{3}+3 x^{2}-3 x+1\right)\).
Short Answer
Expert verified
Answer: The limit of the function as \(x\) approaches \(1\) is 2.
Step by step solution
01
Identify the function and the value of x
The given function is \(f(x) = x^3 + 3x^2 - 3x + 1\), and we are asked to find the limit as \(x \rightarrow 1\).
02
Substitute x with the value it approaches
Since we're finding the limit as \(x\) approaches \(1\), we can substitute \(x\) with \(1\) to get the value of the function at that point: \[f(1) = (1)^3 + 3(1)^2 - 3(1) + 1\]
03
Simplify and evaluate the expression
Now simplify the expression: \[f(1) = 1 + 3 - 3 + 1\] After combining the terms, we get: \[f(1) = 2\]
04
State the final answer
Since the function is continuous at \(x=1\), the limit equals the function's value at that point. Therefore, we have: \[\lim_{x\rightarrow 1} \left(x^3 + 3x^2 - 3x + 1\right) = 2\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Calculus
Calculus is a branch of mathematics that deals with rates of change (differential calculus) and accumulation of quantities (integral calculus). It's a tool that allows us to explore the behavior of functions, understand motion and growth, and solve problems involving complex changes. A fundamental concept in calculus is the limit, which helps us determine what value a function approaches as the input gets closer to a certain number.
When we discuss the limit of a function, we look at the value that a function is approaching as the variable within the function gets infinitely close to a specific point. Limits are crucial to calculus because they give us a way to deal with quantities that are changing at infinitesimally small rates, such as when dealing with continuous functions or the area under a curve.
When we discuss the limit of a function, we look at the value that a function is approaching as the variable within the function gets infinitely close to a specific point. Limits are crucial to calculus because they give us a way to deal with quantities that are changing at infinitesimally small rates, such as when dealing with continuous functions or the area under a curve.
Continuous Functions
Continuous functions are the heart of calculus. They are functions that have no interruptions, jumps, or breaks in their graphs. A function is continuous at a point if the limit of the function as the variable approaches the point is equal to the function's value at that point. This means that we can draw the function's graph at this point without lifting our pen.
A function f(x) is considered continuous at some point c if the limit as x approaches c from both the left and the right exists and matches the actual value of the function at c: \(lim_{x\rightarrow c} f(x) = f(c)\).When a function is continuous over an interval, we can apply the substitution method directly to calculate limits. This is because the behavior of the function around the point matches the behavior at that exact point.
A function f(x) is considered continuous at some point c if the limit as x approaches c from both the left and the right exists and matches the actual value of the function at c: \(lim_{x\rightarrow c} f(x) = f(c)\).When a function is continuous over an interval, we can apply the substitution method directly to calculate limits. This is because the behavior of the function around the point matches the behavior at that exact point.
Limit Evaluation
Evaluating the limit of a function is a key operation in calculus. The limit of a function f(x) as x approaches a value a, denoted \(lim_{x\rightarrow a} f(x)\), is the value that f(x) gets closer to as x approaches a. There are several methods for evaluating limits, including graphical analysis, numerical approximation, algebraic simplification, and, when applicable, direct substitution.
Direct substitution is the most straightforward method when dealing with continuous functions because if the function does not have any gaps or undefined points at x = a, then the limit as x approaches a is simply f(a). The exercise example demonstrates this method by showing the evaluation of a polynomial, which is a type of continuous function, at a specific point to find its limit.
Direct substitution is the most straightforward method when dealing with continuous functions because if the function does not have any gaps or undefined points at x = a, then the limit as x approaches a is simply f(a). The exercise example demonstrates this method by showing the evaluation of a polynomial, which is a type of continuous function, at a specific point to find its limit.
Substitution Method
The substitution method for evaluating limits is a technique used when the function is known to be continuous at the point of interest. In such cases, finding the limit as x approaches a certain value can be as simple as substituting the value of x into the function.
As seen in the provided exercise, after confirming that the polynomial function is continuous at x = 1, the evaluation of the limit involves plugging x = 1 into the function. This method is not only efficient but also highly intuitive because it aligns with our natural understanding of continuous behavior — as the input gets closer to a value, the output gets closer to a corresponding value. Hence, when the function is continuous, the output at the limit is the function's value at that specific input.
As seen in the provided exercise, after confirming that the polynomial function is continuous at x = 1, the evaluation of the limit involves plugging x = 1 into the function. This method is not only efficient but also highly intuitive because it aligns with our natural understanding of continuous behavior — as the input gets closer to a value, the output gets closer to a corresponding value. Hence, when the function is continuous, the output at the limit is the function's value at that specific input.
