Problem 21 - Contemporary Precalculus

Question

Determine whether or not the function is continuous at the given number. $$f(x)=\left\\{\begin{array}{ll} x^{2}-x & \text { if } x \leq 0 \\ 2 x^{2} & \text { if } x>0 \end{array} \text { at } x=0\right.$$

Step-by-Step Solution

Answer

Function: $$ f(x) = \begin{cases} x^2 - x, & \text{if } x \le 0 \\ 2x^2, & \text{if } x > 0 \end{cases} $$ Answer: _Yes_, the function is continuous at x = 0.

Step 1: Evaluate the limit of the function from the left side (x ≤ 0)

Since we are looking at the function from the left side of 0 (x → 0⁻), we will use the definition of the function for x ≤ 0: $$f(x) = x^2 - x$$ Now, let's find the limit as x approaches 0 from the left side: $$\lim_{x \to 0^-} (x^2 - x)$$ Since this is a polynomial function, we can simply substitute x = 0: $$\lim_{x \to 0^-} (x^2 - x) = (0)^2 - 0 = 0$$ The limit from the left side is 0.

Step 2: Evaluate the limit of the function from the right side (x > 0)

Now, we will look at the function from the right side of 0 (x → 0⁺). In this case, we will use the definition of the function for x > 0: $$f(x) = 2x^2$$ Let's find the limit as x approaches 0 from the right side: $$\lim_{x \to 0^+} (2x^2)$$ This is also a polynomial function, so we can simply substitute x = 0: $$\lim_{x \to 0^+} (2x^2) = 2(0)^2 = 0$$ The limit from the right side is 0.

Step 3: Compare the limits and the function value at x = 0

Now that we have the limits from both sides, we can check if they are equal and if they equal the function value at x = 0. From our calculations: $$\lim_{x \to 0^-} f(x) = 0$$$$\lim_{x \to 0^+} f(x) = 0$$ and $$f(0) = 0^2 - 0 = 0$$ Since the limits from both sides are equal and equal to the function value at x = 0, the function is continuous at x = 0.

Key Concepts

Limits

Limits are a fundamental concept in calculus used to describe the behavior of a function as its input approaches a certain value. In simpler terms, it helps us understand what happens to the function values when the input gets very close to a specific point.

For example, suppose we have a function, and we want to know what happens to the function as the input (\( x \)) gets close to 0. We look at the function values as \( x \) approaches 0 from both the left (\( x \to 0^- \)) and the right (\( x \to 0^+ \)). These are known as one-sided limits. If both limits exist and are equal, the overall limit at that point also exists.

• One-sided limit from the left: Given by \( \lim_{x \to a^-} f(x) \)
• One-sided limit from the right: Given by \( \lim_{x \to a^+} f(x) \)

If both one-sided limits are the same, we write \( \lim_{x \to a} f(x) = L \).
Understanding limits is crucial when determining the continuity of a function at a particular point.

Piecewise Functions

Piecewise functions are mathematical functions defined by different expressions for different intervals of the input. They are "piecewise" because each part of the function applies only over a certain range of the variable.
Here's how you understand piecewise functions:

• They consist of multiple sub-functions, each with its own formula.
• Each sub-function applies to a specific interval or condition.
• The function value will change based on the interval or condition that \( x \) falls into.

For instance, in this exercise, the function \( f(x) \) is defined as:

• \( x^2 - x \) for \( x \leq 0 \)
• \( 2x^2 \) for \( x > 0 \)

This means the function changes its formula depending on whether \( x \) is less than or equal to or greater than 0. Evaluating limits and continuity often involves checking each piece separately to ensure the function behaves consistently across boundaries.

Polynomial Functions

Polynomial functions are among the most straightforward types of functions in mathematics and are composed of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Examples include linear functions like \( f(x) = x \), quadratic functions like \( f(x) = x^2 \), and more complex polynomials like \( f(x) = x^3 - 4x + 5 \).
Some key features of polynomial functions include:

• They are continuous for all real numbers, meaning there are no breaks, jumps, or holes in their graphs.
• They are defined for all values of \( x \) from negative to positive infinity.
• Polynomial functions retain smooth and predictable shapes, such as lines, parabolas, and curves.

In this exercise, both pieces of the piecewise function are polynomial functions, \( x^2 - x \) and \( 2x^2 \). This property makes evaluating limits straightforward because polynomial functions are always continuous wherever they are defined. We can simply substitute the value of \( x \) directly to find limits, as done in the solution.
Understanding the nature of polynomial functions helps in grasping continuity and evaluating limits when dealing with functions made of polynomials.