91Ó°ÊÓ

Continuity is a fundamental property of functions that describes how smoothly they behave. A function is continuous at a point if there is no interruption in its graph at that point. Mathematically, a function \( f(x) \) is continuous at \( x = c \) if three conditions are met:

• \( f(c) \) is defined, meaning the function exists at \( x = c \).
• The limit \( \lim_{x \to c} f(x) \) exists.
• \( \lim_{x \to c} f(x) = f(c) \).

For polynomials, which will be discussed later, continuity is easy to guarantee since they are smooth everywhere. However, when a polynomial is inside a square root, as in our exercise, continuity depends not only on the polynomial being defined but also on the square root being real.
To ensure this, the expression inside the square root must remain non-negative around the point of interest. That's why confirming the non-negativity of the polynomial \( x^4 - 4x + 1 \) at \( x = -2 \) is crucial for establishing continuity.

Polynomials are algebraic expressions consisting of variables raised to whole number powers with constant coefficients. Examples include \( x^2 + 3x + 5 \) or \( x^4 - 4x + 1 \). These expressions are known for their predictable and smooth curve-like graphs.

A key feature of polynomials is their continuity across all real numbers. This means, wherever you plug in an \( x \) value, the polynomial will smoothly map this input to a corresponding output on its graph without any breaks or jumps.

• Polynomials like the one in the exercise, \( x^4 - 4x + 1 \), can take any real number as input.
• This polynomial specifically is a fourth-degree polynomial which ensures it will always be continuous for all real values of \( x \).

Given the continuity characteristics of polynomials, one can easily apply limit laws to handle problems that involve limits and polynomials. It's through this smooth behavior that tasks involving polynomial limits become straightforward because \( \lim_{x \to c} g(x) = g(c) \) when \( g(x) \) is a polynomial.

Square root functions involve taking the square root of an expression and they present unique challenges due to their dependency on non-negative inputs. This is because square roots of negative numbers are not real in standard math, hence a square root function is only real and defined when its input is zero or positive.

For a square root function like \( \sqrt{g(x)} \), where \( g(x) \) is a polynomial, the output will be continuous wherever \( g(x) \) is non-negative, as long as \( g(x) \) is itself continuous.

• Since polynomials are universally continuous, the main concern becomes ensuring \( g(x) \) remains non-negative in the range considered.
• In our exercise's example, the polynomial \( x^4 - 4x + 1 \) must be calculated at \( x = -2 \) to check if it yields a non-negative result.

By substituting \( -2 \) into the polynomial, we verified it returns a positive 25, thus, the square root function \( \sqrt{x^4 - 4x + 1} \) is continuous at \( x = -2 \). This allows us to use the limit theorem directly, proving the existence of the limit.