Problem 12 - Calculus and its applications

Question

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow-2}\left(x^{2}+3\right) $$

Step-by-Step Solution

Answer

The limit is 7.

Step 1: Identify the Type of Function

The function given is a polynomial, specifically a quadratic function since it is composed of \( x^2 \) and constant terms.

Step 2: Apply the Theorem on Limits of Rational Functions

The Theorem on Limits for polynomials states that if \( f(x) \) is a polynomial, then \( \lim_{x \to a} f(x) = f(a) \) for any real number \( a \). Polynomials are continuous everywhere, meaning we can directly substitute the value of \( x \).

Step 3: Substitute \( x = -2 \) into the Polynomial

Substitute \( x = -2 \) into the function \( f(x) = x^2 + 3 \):\[ f(-2) = (-2)^2 + 3 = 4 + 3 = 7 \]

Step 4: Conclusion

Since the polynomial \( f(x) \) is continuous at \( x = -2 \), the limit is equal to the value of the function at that point, which is 7.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to whole number powers and multiplied by coefficients. These functions can take various forms, and the degree of a polynomial indicates the highest power of the variable present. Some key characteristics of polynomial functions include:

• They are composed of terms like \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer.
• The polynomial is named after its highest degree, such as quadratic for degree 2, cubic for degree 3, and so on.
• Polynomial functions include linear functions (degree 1), quadratic functions (degree 2), and so forth.
• For example, \( f(x) = x^2 + 3 \) is a quadratic polynomial function.

Polynomial functions are a fundamental part of algebra and calculus because they model a wide variety of real-life situations.

Continuous Functions

A function is continuous if, at any given point in its domain, there are no breaks, jumps, or holes. Understanding continuity helps in analyzing limits and behavior of functions:

• A continuous function can be graphed as a single unbroken line.
• If a function is continuous at a specific point \( a \), the limit of the function as \( x \) approaches \( a \) is equal to the function's value at \( a \).
  In our exercise, since \( f(x) = x^2 + 3 \) is continuous everywhere, we can simply evaluate the function at the point \( x = -2 \) to find its limit.
• Continuity is an essential property that simplifies evaluating limits, especially for polynomial functions.

Continuous functions ensure that a small change in input results in a small change in output, maintaining stability and predictability.

Limit Theorem for Polynomials

The Limit Theorem for Polynomials provides a straightforward approach to finding limits for polynomial functions.According to this theorem, if you have a polynomial function \( f(x) \), then:

• The limit of \( f(x) \) as \( x \) approaches any real number \( a \) is simply the value of the polynomial at that point: \[ \lim_{x \to a} f(x) = f(a) \]
• This theorem is applicable because polynomial functions are continuous across their entire domain.
• For instance, in the given exercise, the limit of \( f(x) = x^2 + 3 \) as \( x \) approaches \(-2\) can be found by direct substitution: \[ f(-2) = (-2)^2 + 3 = 7 \]
• No matter the degree of the polynomial, as long as it is finite and its domain encompasses the point \( a \), the theorem holds true.

This theorem enables quick and efficient calculation of limits without the need for complex algebraic manipulation.