Problem 11 - Calculus Early Transcendentals

Question

\(10-12\) Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. \(f(x)=\left(x+2 x^{3}\right)^{4}, \quad a=-1\)

Step-by-Step Solution

Answer

The function is continuous at \( a = -1 \).

Step 1: Recall the Definition of Continuity

A function \( f(x) \) is continuous at a point \( a \) if three conditions are met: \( f(a) \) is defined, the limit of \( f(x) \) as \( x \to a \) exists, and the limit of \( f(x) \) as \( x \to a \) equals \( f(a) \).

Step 2: Show that \( f(a) \) is Defined

Substitute \( a = -1 \) into the function:\[ f(-1) = ((-1) + 2(-1)^{3})^{4} = (-1 - 2)^{4} = (-3)^{4} = 81. \]Thus, \( f(-1) \) is defined and equals 81.

Step 3: Calculate the Limit as \( x \to a \)

Find \( \lim_{x \to -1} f(x) \) by substituting into the function:\[ \lim_{x \to -1} (x + 2x^3)^4 = \lim_{x \to -1} ((x + 2x^3))^4. \]As polynomials are continuous, this equals:\[ ((-1) + 2(-1)^3)^4 = (-1 - 2)^4 = (-3)^4 = 81. \]Thus, the limit exists and is 81.

Step 4: Compare \( f(a) \) and the Limit

Since \( \lim_{x \to -1} f(x) = 81 \) and \( f(-1) = 81 \), we conclude that \( \lim_{x \to -1} f(x) = f(-1) \). Therefore, the function is continuous at \( a = -1 \).

Key Concepts

Definition of continuity

Continuity is an essential concept in calculus that helps to determine the behavior of functions at specific points. A function \( f(x) \) is declared to be continuous at a point \( a \) if all the following conditions are fulfilled:

• The function value \( f(a) \) exists. This means \( a \) is within the domain of the function.
• The limit \( \lim_{x \to a} f(x) \) exists. The limit is the value that \( f(x) \) approaches as \( x \) gets closer to \( a \).
• The limit of \( f(x) \) as \( x \) approaches \( a \) equals \( f(a) \). In mathematical terms, \( \lim_{x \to a} f(x) = f(a) \).

By verifying all these points at a particular \( a \), you can conclude if the function is continuous there. Continuity ensures the function behaves predictably without jumps or abrupt changes when near \( a \). This helps in understanding and predicting how the function will behave elsewhere.

Properties of limits

Understanding the properties of limits is crucial when discussing continuity since they directly impact it. Limits of functions describe how functions behave as inputs approach a specific value. Here are some key properties of limits that are useful in showing continuity:

• Limit of a Sum: \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \). Limits can be distributed over addition.
• Limit of a Product: \( \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \). Though both limits must exist individually first.
• Limit of a Power: For any integer \( n \), \( \lim_{x \to a} [f(x)]^n = (\lim_{x \to a} f(x))^n \). This is fundamental for dealing with polynomial functions.

These properties allow us to manipulate and simplify the expressions involving limits. By understanding these rules, you can more easily determine if a function is continuous at a point using the definition of continuity.

Polynomials continuity

Polynomials are among the most straightforward functions in terms of continuity. A polynomial function is one composed of terms where each consists of a variable raised to a power and multiplied by a coefficient. Examples include \( f(x) = x^2 + 3x + 5 \). Such functions have some consistent properties that make them inherently continuous:

• Smooth and Unbroken: Polynomials are always smooth and continuous across their entire domain. There are no gaps, jumps, or asymptotes.
• Continuous Everywhere: Because they are continuous everywhere by nature, polynomials simplify our work when checking for limits and continuity at specific points. We only need to check the function value, knowing the limit will match the value.
• Usefulness in Calculus: This property of being continuous everywhere makes polynomials straightforward to differentiate and integrate, crucial operations in calculus.

When proving the continuity of a polynomial function at a point, the focus can be on confirming that it satisfies the definition of continuity, due to their intrinsic properties.