Problem 29 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated limit. \(\lim _{x \rightarrow 1}\left(2 x^{3}-3 x^{2}+x+2\right)\)

Step-by-Step Solution

Answer

The short answer is: \(\lim_{x \rightarrow 1} (2x^3 - 3x^2 + x + 2) = 2\).

Step 1: Identify the limit

As x approaches 1, we need to find the limit of the function \(2x^3 - 3x^2 + x + 2\). This function is a polynomial, so we can directly substitute x=1 into the function to find its limit.

Step 2: Substitute x=1 into the function

Replace x by 1 in the given function \(2x^3 - 3x^2 + x + 2\): \(2(1)^3 - 3(1)^2 + (1) + 2\)

Step 3: Evaluate the function

Now, we just need to perform the arithmetic operations: \(2(1) - 3(1) + 1 + 2\) \((2 - 3 + 1 + 2)\)

Step 4: Simplify the expression

Combine like terms: \(-1 + 1 + 2\) \(0 + 2 = 2\)

Step 5: State the limit

The limit of the function \(2x^3 - 3x^2 + x + 2\) as x approaches 1 is 2: \(\lim_{x \rightarrow 1} (2x^3 - 3x^2 + x + 2) = 2\)

Key Concepts

Limits

In calculus, limits help us understand how a function behaves as the input approaches a certain value. They're essential in analyzing functions and building the foundation for derivatives and integrals.
To find a limit, such as \(\lim_{x \to 1} (2x^3 - 3x^2 + x + 2)\), you observe the behavior of the function as x gets closer to a specific value, which is 1 in this case.

• Limits can sometimes reveal the value the function approaches, even if it doesn't reach that value exactly.
• For polynomials, like our given function, finding limits is often straightforward because they are continuous.

Understanding limits is crucial for working with more complex calculus concepts, such as continuity and differentiability. They essentially measure the tendency of a sequence or function as it approaches a particular point.
Calculating limits for polynomials often involves direct substitution, which we'll explore in the next sections.

Polynomials

Polynomials are mathematical expressions that involve variables raised to whole number exponents. They can have multiple terms, like the example given by \(2x^3 - 3x^2 + x + 2\). Each term can have:

• A coefficient (like 2, -3, or 1 in this case)
• A variable (in this case, x)
• An exponent (which can be 3, 2, or 1 in these terms)

Polynomials are one of the most common types of functions you'll encounter in algebra and calculus. They are particularly nice to work with because they are continuous and differentiable everywhere.
In this limit problem, because our function is a polynomial, we can confidently apply the substitution method to find the limit. Polynomials allow us to use direct substitution due to their continuity properties.

Substitution Method

The substitution method is a simple and effective way to calculate limits, especially with polynomial functions. This method involves directly replacing the variable with the given value.
In our exercise, we substituted \(x = 1\) into the polynomial \(2x^3 - 3x^2 + x + 2\), allowing us to simplify the expression step by step:

• Starting by replacing x with 1: \(2(1)^3 - 3(1)^2 + (1) + 2\)
• Calculating each part: \(2(1) - 3(1) + 1 + 2\)
• Combining like terms: \(-1 + 1 + 2\)
• Simplifying to reach the value of 2

This method is straightforward because polynomial functions are well-behaved, and as such, the limit can often be found by simple substitution. Direct substitution stands out for its simplicity and efficiency when dealing with these types of expressions.